| L(s) = 1 | − 7·3-s − 3·5-s − 4·7-s + 56·9-s − 3·11-s + 52·13-s + 21·15-s + 31·17-s − 89·19-s + 28·21-s + 201·23-s + 42·25-s − 91·27-s − 380·29-s + 339·31-s + 21·33-s + 12·35-s − 535·37-s − 364·39-s + 116·41-s − 536·43-s − 168·45-s + 205·47-s − 757·49-s − 217·51-s + 757·53-s + 9·55-s + ⋯ |
| L(s) = 1 | − 1.34·3-s − 0.268·5-s − 0.215·7-s + 2.07·9-s − 0.0822·11-s + 1.10·13-s + 0.361·15-s + 0.442·17-s − 1.07·19-s + 0.290·21-s + 1.82·23-s + 0.335·25-s − 0.648·27-s − 2.43·29-s + 1.96·31-s + 0.110·33-s + 0.0579·35-s − 2.37·37-s − 1.49·39-s + 0.441·41-s − 1.90·43-s − 0.556·45-s + 0.636·47-s − 2.20·49-s − 0.595·51-s + 1.96·53-s + 0.0220·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.961704259\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.961704259\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 4 T + 773 T^{2} + 328 p T^{3} + 773 p^{3} T^{4} + 4 p^{6} T^{5} + p^{9} T^{6} \) |
| good | 3 | \( 1 + 7 T - 7 T^{2} - 350 T^{3} - 973 T^{4} + 4711 T^{5} + 56614 T^{6} + 4711 p^{3} T^{7} - 973 p^{6} T^{8} - 350 p^{9} T^{9} - 7 p^{12} T^{10} + 7 p^{15} T^{11} + p^{18} T^{12} \) |
| 5 | \( 1 + 3 T - 33 T^{2} + 2752 T^{3} + 453 T^{4} - 39771 T^{5} + 5269094 T^{6} - 39771 p^{3} T^{7} + 453 p^{6} T^{8} + 2752 p^{9} T^{9} - 33 p^{12} T^{10} + 3 p^{15} T^{11} + p^{18} T^{12} \) |
| 11 | \( 1 + 3 T - 1671 T^{2} - 88206 T^{3} + 446451 T^{4} + 70224843 T^{5} + 3117041062 T^{6} + 70224843 p^{3} T^{7} + 446451 p^{6} T^{8} - 88206 p^{9} T^{9} - 1671 p^{12} T^{10} + 3 p^{15} T^{11} + p^{18} T^{12} \) |
| 13 | \( ( 1 - 2 p T + 4347 T^{2} - 140572 T^{3} + 4347 p^{3} T^{4} - 2 p^{7} T^{5} + p^{9} T^{6} )^{2} \) |
| 17 | \( 1 - 31 T - 7341 T^{2} + 5924 p T^{3} + 22585393 T^{4} + 294225803 T^{5} - 92586620842 T^{6} + 294225803 p^{3} T^{7} + 22585393 p^{6} T^{8} + 5924 p^{10} T^{9} - 7341 p^{12} T^{10} - 31 p^{15} T^{11} + p^{18} T^{12} \) |
| 19 | \( 1 + 89 T - 13039 T^{2} - 531306 T^{3} + 188901763 T^{4} + 3714875569 T^{5} - 1304484035114 T^{6} + 3714875569 p^{3} T^{7} + 188901763 p^{6} T^{8} - 531306 p^{9} T^{9} - 13039 p^{12} T^{10} + 89 p^{15} T^{11} + p^{18} T^{12} \) |
| 23 | \( 1 - 201 T - 7035 T^{2} + 419006 T^{3} + 672679263 T^{4} - 33819862221 T^{5} - 4987613588554 T^{6} - 33819862221 p^{3} T^{7} + 672679263 p^{6} T^{8} + 419006 p^{9} T^{9} - 7035 p^{12} T^{10} - 201 p^{15} T^{11} + p^{18} T^{12} \) |
| 29 | \( ( 1 + 190 T + 81643 T^{2} + 9219316 T^{3} + 81643 p^{3} T^{4} + 190 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 31 | \( 1 - 339 T + 14469 T^{2} + 1234882 T^{3} + 1116704127 T^{4} - 44284930023 T^{5} - 38410082180298 T^{6} - 44284930023 p^{3} T^{7} + 1116704127 p^{6} T^{8} + 1234882 p^{9} T^{9} + 14469 p^{12} T^{10} - 339 p^{15} T^{11} + p^{18} T^{12} \) |
| 37 | \( 1 + 535 T + 77743 T^{2} + 4911224 T^{3} + 118429129 p T^{4} + 1188076376281 T^{5} + 166239153232886 T^{6} + 1188076376281 p^{3} T^{7} + 118429129 p^{7} T^{8} + 4911224 p^{9} T^{9} + 77743 p^{12} T^{10} + 535 p^{15} T^{11} + p^{18} T^{12} \) |
| 41 | \( ( 1 - 58 T + 164327 T^{2} - 4134444 T^{3} + 164327 p^{3} T^{4} - 58 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 43 | \( ( 1 + 268 T + 132841 T^{2} + 18272264 T^{3} + 132841 p^{3} T^{4} + 268 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 47 | \( 1 - 205 T - 183403 T^{2} + 16447246 T^{3} + 20990523375 T^{4} + 323493805591 T^{5} - 2509221820128074 T^{6} + 323493805591 p^{3} T^{7} + 20990523375 p^{6} T^{8} + 16447246 p^{9} T^{9} - 183403 p^{12} T^{10} - 205 p^{15} T^{11} + p^{18} T^{12} \) |
| 53 | \( 1 - 757 T + 229743 T^{2} + 41591296 T^{3} - 47359243259 T^{4} + 8853768931901 T^{5} - 926599751279866 T^{6} + 8853768931901 p^{3} T^{7} - 47359243259 p^{6} T^{8} + 41591296 p^{9} T^{9} + 229743 p^{12} T^{10} - 757 p^{15} T^{11} + p^{18} T^{12} \) |
| 59 | \( 1 + 1799 T + 1576225 T^{2} + 1115416106 T^{3} + 732038218331 T^{4} + 402383977564751 T^{5} + 190135753613171014 T^{6} + 402383977564751 p^{3} T^{7} + 732038218331 p^{6} T^{8} + 1115416106 p^{9} T^{9} + 1576225 p^{12} T^{10} + 1799 p^{15} T^{11} + p^{18} T^{12} \) |
| 61 | \( 1 - 625 T - 246217 T^{2} + 136351344 T^{3} + 94023700837 T^{4} - 21668489332199 T^{5} - 17403170402691818 T^{6} - 21668489332199 p^{3} T^{7} + 94023700837 p^{6} T^{8} + 136351344 p^{9} T^{9} - 246217 p^{12} T^{10} - 625 p^{15} T^{11} + p^{18} T^{12} \) |
| 67 | \( 1 + 495 T - 430551 T^{2} - 36746478 T^{3} + 161353409283 T^{4} - 31007237744241 T^{5} - 70581306208381018 T^{6} - 31007237744241 p^{3} T^{7} + 161353409283 p^{6} T^{8} - 36746478 p^{9} T^{9} - 430551 p^{12} T^{10} + 495 p^{15} T^{11} + p^{18} T^{12} \) |
| 71 | \( ( 1 - 640 T + 1196037 T^{2} - 465417472 T^{3} + 1196037 p^{3} T^{4} - 640 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( 1 - 443 T - 406693 T^{2} + 622914612 T^{3} - 68154601487 T^{4} - 127540200423001 T^{5} + 144654083731272358 T^{6} - 127540200423001 p^{3} T^{7} - 68154601487 p^{6} T^{8} + 622914612 p^{9} T^{9} - 406693 p^{12} T^{10} - 443 p^{15} T^{11} + p^{18} T^{12} \) |
| 79 | \( 1 + p T - 940435 T^{2} + 213350022 T^{3} + 434920484023 T^{4} - 120129257254237 T^{5} - 198873325035594506 T^{6} - 120129257254237 p^{3} T^{7} + 434920484023 p^{6} T^{8} + 213350022 p^{9} T^{9} - 940435 p^{12} T^{10} + p^{16} T^{11} + p^{18} T^{12} \) |
| 83 | \( ( 1 - 2372 T + 3356817 T^{2} - 2979344792 T^{3} + 3356817 p^{3} T^{4} - 2372 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 89 | \( 1 + 821 T - 895221 T^{2} - 833200796 T^{3} + 485399771137 T^{4} + 256153722115943 T^{5} - 245058388113291418 T^{6} + 256153722115943 p^{3} T^{7} + 485399771137 p^{6} T^{8} - 833200796 p^{9} T^{9} - 895221 p^{12} T^{10} + 821 p^{15} T^{11} + p^{18} T^{12} \) |
| 97 | \( ( 1 + 342 T + 11679 T^{2} + 841589780 T^{3} + 11679 p^{3} T^{4} + 342 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.67477483600321017438985428283, −5.18631218294110848349046503401, −5.01242876688093585355879539575, −4.93248694890728635556496067775, −4.82677311773140415896619401721, −4.80702705644141529297265376286, −4.72133954525533282089212281347, −4.19996289360554350656073417885, −4.05034970713377870312402110295, −3.86345278712985361287010084113, −3.66030146026013192656131967534, −3.47328753435172540415278059969, −3.27291564245473603064669454160, −3.18356488266125512763472656157, −2.89895318812325700315437532636, −2.65139779361766959150216945248, −2.12885061169133961700729543221, −2.00894443011009932089532117341, −1.80522746020129402725252227013, −1.62987688819571003712722124716, −1.29676463747758517786720818996, −0.865024692494759317481763741351, −0.78865202048150009922617986408, −0.56819401759378883903723423821, −0.21958077998283339074240728515,
0.21958077998283339074240728515, 0.56819401759378883903723423821, 0.78865202048150009922617986408, 0.865024692494759317481763741351, 1.29676463747758517786720818996, 1.62987688819571003712722124716, 1.80522746020129402725252227013, 2.00894443011009932089532117341, 2.12885061169133961700729543221, 2.65139779361766959150216945248, 2.89895318812325700315437532636, 3.18356488266125512763472656157, 3.27291564245473603064669454160, 3.47328753435172540415278059969, 3.66030146026013192656131967534, 3.86345278712985361287010084113, 4.05034970713377870312402110295, 4.19996289360554350656073417885, 4.72133954525533282089212281347, 4.80702705644141529297265376286, 4.82677311773140415896619401721, 4.93248694890728635556496067775, 5.01242876688093585355879539575, 5.18631218294110848349046503401, 5.67477483600321017438985428283
Plot not available for L-functions of degree greater than 10.
