| L(s) = 1 | + 3-s + 3·5-s + 7-s + 3·9-s − 7·13-s + 3·15-s + 3·17-s − 7·19-s + 21-s + 8·23-s + 18·25-s + 3·35-s + 4·37-s − 7·39-s − 17·41-s + 16·43-s + 9·45-s + 16·47-s + 5·49-s + 3·51-s − 19·53-s − 7·57-s − 9·59-s + 7·61-s + 3·63-s − 21·65-s + 9·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 0.377·7-s + 9-s − 1.94·13-s + 0.774·15-s + 0.727·17-s − 1.60·19-s + 0.218·21-s + 1.66·23-s + 18/5·25-s + 0.507·35-s + 0.657·37-s − 1.12·39-s − 2.65·41-s + 2.43·43-s + 1.34·45-s + 2.33·47-s + 5/7·49-s + 0.420·51-s − 2.60·53-s − 0.927·57-s − 1.17·59-s + 0.896·61-s + 0.377·63-s − 2.60·65-s + 1.09·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.855916866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.855916866\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 - 4 T + 92 T^{2} - 238 T^{3} + 92 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 3 | \( 1 - T - 2 T^{2} + 5 T^{3} - 4 T^{4} - p T^{5} + 34 T^{6} - p^{2} T^{7} - 4 p^{2} T^{8} + 5 p^{3} T^{9} - 2 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) |
| 5 | \( ( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} )^{3} \) |
| 7 | \( 1 - T - 4 T^{2} - T^{3} - 12 T^{4} + 11 T^{5} + 466 T^{6} + 11 p T^{7} - 12 p^{2} T^{8} - p^{3} T^{9} - 4 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( ( 1 + 19 T^{2} + 16 T^{3} + 19 p T^{4} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 + 7 T + 18 T^{2} - 3 T^{3} - 212 T^{4} - 1049 T^{5} - 4280 T^{6} - 1049 p T^{7} - 212 p^{2} T^{8} - 3 p^{3} T^{9} + 18 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 - 3 T + 11 T^{2} - 156 T^{3} + 281 T^{4} - 1233 T^{5} + 13718 T^{6} - 1233 p T^{7} + 281 p^{2} T^{8} - 156 p^{3} T^{9} + 11 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 + 7 T - 18 T^{2} - 59 T^{3} + 72 p T^{4} + 2393 T^{5} - 19566 T^{6} + 2393 p T^{7} + 72 p^{3} T^{8} - 59 p^{3} T^{9} - 18 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( ( 1 - 4 T + 43 T^{2} - 88 T^{3} + 43 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 + 28 T^{2} - 6 p T^{3} + 28 p T^{4} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 79 T^{2} - 16 T^{3} + 79 p T^{4} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 + 17 T + 95 T^{2} + 432 T^{3} + 5321 T^{4} + 36143 T^{5} + 162446 T^{6} + 36143 p T^{7} + 5321 p^{2} T^{8} + 432 p^{3} T^{9} + 95 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 8 T + 85 T^{2} - 480 T^{3} + 85 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( ( 1 - 8 T + 63 T^{2} - 160 T^{3} + 63 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( 1 + 19 T + 98 T^{2} + 681 T^{3} + 14228 T^{4} + 91243 T^{5} + 274712 T^{6} + 91243 p T^{7} + 14228 p^{2} T^{8} + 681 p^{3} T^{9} + 98 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 + 9 T - 40 T^{2} - 43 T^{3} + 2408 T^{4} - 28687 T^{5} - 414666 T^{6} - 28687 p T^{7} + 2408 p^{2} T^{8} - 43 p^{3} T^{9} - 40 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 7 T - 125 T^{2} + 396 T^{3} + 14013 T^{4} - 17869 T^{5} - 923954 T^{6} - 17869 p T^{7} + 14013 p^{2} T^{8} + 396 p^{3} T^{9} - 125 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 9 T - 10 T^{2} - 291 T^{3} - 192 T^{4} + 45321 T^{5} - 247214 T^{6} + 45321 p T^{7} - 192 p^{2} T^{8} - 291 p^{3} T^{9} - 10 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 + 15 T + 20 T^{2} - 57 T^{3} + 3980 T^{4} - 40005 T^{5} - 949822 T^{6} - 40005 p T^{7} + 3980 p^{2} T^{8} - 57 p^{3} T^{9} + 20 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( ( 1 - 22 T + 355 T^{2} - 3420 T^{3} + 355 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( 1 + 15 T - 30 T^{2} - 591 T^{3} + 11760 T^{4} + 52305 T^{5} - 679858 T^{6} + 52305 p T^{7} + 11760 p^{2} T^{8} - 591 p^{3} T^{9} - 30 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 - 15 T + 38 T^{2} - 61 T^{3} - 1132 T^{4} + 105731 T^{5} - 1426494 T^{6} + 105731 p T^{7} - 1132 p^{2} T^{8} - 61 p^{3} T^{9} + 38 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 + 17 T + 27 T^{2} - 804 T^{3} + 633 T^{4} + 30643 T^{5} - 76906 T^{6} + 30643 p T^{7} + 633 p^{2} T^{8} - 804 p^{3} T^{9} + 27 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( ( 1 + 4 T + 104 T^{2} + 1250 T^{3} + 104 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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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
−5.93167298189212338003965530543, −5.36071449538516443739461965114, −5.25348923935079515668957889673, −5.15549569415051409360589557828, −5.12469673524202706910202323513, −4.80002108878106255396411323030, −4.69752054207521991822271997731, −4.64485844093311378852595170605, −4.50572119382934479572862400354, −4.24682874344543763245977686977, −3.79475800993879104745308891001, −3.61585343155879512729226018716, −3.61563060868960577655738816220, −3.45948987602340826279893889791, −3.03603364640843091421160148765, −2.82808549543258587843573822713, −2.63171120006187449469576069294, −2.41405817902743444667782395391, −2.29721989519170572514837296450, −2.09130553950346305078819700985, −1.96239056929688746112637619044, −1.34965228698998507930127492707, −1.22648081865298083817727439610, −1.00121999316913041046562763312, −0.48368904500265862225070248081,
0.48368904500265862225070248081, 1.00121999316913041046562763312, 1.22648081865298083817727439610, 1.34965228698998507930127492707, 1.96239056929688746112637619044, 2.09130553950346305078819700985, 2.29721989519170572514837296450, 2.41405817902743444667782395391, 2.63171120006187449469576069294, 2.82808549543258587843573822713, 3.03603364640843091421160148765, 3.45948987602340826279893889791, 3.61563060868960577655738816220, 3.61585343155879512729226018716, 3.79475800993879104745308891001, 4.24682874344543763245977686977, 4.50572119382934479572862400354, 4.64485844093311378852595170605, 4.69752054207521991822271997731, 4.80002108878106255396411323030, 5.12469673524202706910202323513, 5.15549569415051409360589557828, 5.25348923935079515668957889673, 5.36071449538516443739461965114, 5.93167298189212338003965530543
Plot not available for L-functions of degree greater than 10.
