| L(s) = 1 | + 8·3-s + 10·9-s + 24·11-s − 56·23-s − 88·27-s + 20·31-s + 192·33-s − 72·37-s + 184·47-s + 74·49-s − 136·53-s − 16·59-s + 264·67-s − 448·69-s − 220·71-s − 389·81-s + 220·89-s + 160·93-s + 8·97-s + 240·99-s + 80·103-s − 576·111-s − 784·113-s + 48·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 8/3·3-s + 10/9·9-s + 2.18·11-s − 2.43·23-s − 3.25·27-s + 0.645·31-s + 5.81·33-s − 1.94·37-s + 3.91·47-s + 1.51·49-s − 2.56·53-s − 0.271·59-s + 3.94·67-s − 6.49·69-s − 3.09·71-s − 4.80·81-s + 2.47·89-s + 1.72·93-s + 8/97·97-s + 2.42·99-s + 0.776·103-s − 5.18·111-s − 6.93·113-s + 0.396·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(18.21621435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.21621435\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 24 T + 48 p T^{2} - 7752 T^{3} + 8634 p T^{4} - 7752 p^{2} T^{5} + 48 p^{5} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| good | 3 | \( ( 1 - 4 T + 19 T^{2} - 56 T^{3} + 224 T^{4} - 56 p^{2} T^{5} + 19 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 7 | \( 1 - 74 T^{2} + 3977 T^{4} - 167750 T^{6} + 3500796 T^{8} - 167750 p^{4} T^{10} + 3977 p^{8} T^{12} - 74 p^{12} T^{14} + p^{16} T^{16} \) |
| 13 | \( 1 - 300 T^{2} + 81236 T^{4} - 18352980 T^{6} + 2938128406 T^{8} - 18352980 p^{4} T^{10} + 81236 p^{8} T^{12} - 300 p^{12} T^{14} + p^{16} T^{16} \) |
| 17 | \( 1 - 434 T^{2} + 149937 T^{4} - 22636510 T^{6} + 7111449836 T^{8} - 22636510 p^{4} T^{10} + 149937 p^{8} T^{12} - 434 p^{12} T^{14} + p^{16} T^{16} \) |
| 19 | \( 1 - 2238 T^{2} + 2379913 T^{4} - 1554252886 T^{6} + 678586096820 T^{8} - 1554252886 p^{4} T^{10} + 2379913 p^{8} T^{12} - 2238 p^{12} T^{14} + p^{16} T^{16} \) |
| 23 | \( ( 1 + 28 T + 80 p T^{2} + 39428 T^{3} + 1386334 T^{4} + 39428 p^{2} T^{5} + 80 p^{5} T^{6} + 28 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 29 | \( 1 - 94 p T^{2} + 4239617 T^{4} - 4657885310 T^{6} + 4180681051236 T^{8} - 4657885310 p^{4} T^{10} + 4239617 p^{8} T^{12} - 94 p^{13} T^{14} + p^{16} T^{16} \) |
| 31 | \( ( 1 - 10 T + 1077 T^{2} - 23990 T^{3} + 1285668 T^{4} - 23990 p^{2} T^{5} + 1077 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 + 36 T + 4655 T^{2} + 121852 T^{3} + 8960448 T^{4} + 121852 p^{2} T^{5} + 4655 p^{4} T^{6} + 36 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( 1 - 4248 T^{2} + 10605308 T^{4} - 15356624296 T^{6} + 23325534737670 T^{8} - 15356624296 p^{4} T^{10} + 10605308 p^{8} T^{12} - 4248 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( 1 - 9220 T^{2} + 42113396 T^{4} - 124400856060 T^{6} + 266109939612246 T^{8} - 124400856060 p^{4} T^{10} + 42113396 p^{8} T^{12} - 9220 p^{12} T^{14} + p^{16} T^{16} \) |
| 47 | \( ( 1 - 92 T + 10112 T^{2} - 611044 T^{3} + 34913854 T^{4} - 611044 p^{2} T^{5} + 10112 p^{4} T^{6} - 92 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 + 68 T + 10343 T^{2} + 553620 T^{3} + 42507456 T^{4} + 553620 p^{2} T^{5} + 10343 p^{4} T^{6} + 68 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + 8 T + 9160 T^{2} - 63416 T^{3} + 38801518 T^{4} - 63416 p^{2} T^{5} + 9160 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 61 | \( 1 - 12646 T^{2} + 92928177 T^{4} - 473273403470 T^{6} + 1944383189603876 T^{8} - 473273403470 p^{4} T^{10} + 92928177 p^{8} T^{12} - 12646 p^{12} T^{14} + p^{16} T^{16} \) |
| 67 | \( ( 1 - 132 T + 13748 T^{2} - 1229340 T^{3} + 93932086 T^{4} - 1229340 p^{2} T^{5} + 13748 p^{4} T^{6} - 132 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 + 110 T + 13837 T^{2} + 684750 T^{3} + 63811148 T^{4} + 684750 p^{2} T^{5} + 13837 p^{4} T^{6} + 110 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 73 | \( 1 - 14700 T^{2} + 92426196 T^{4} - 609327092820 T^{6} + 4081053740259926 T^{8} - 609327092820 p^{4} T^{10} + 92426196 p^{8} T^{12} - 14700 p^{12} T^{14} + p^{16} T^{16} \) |
| 79 | \( 1 - 19480 T^{2} + 289539196 T^{4} - 2648556220200 T^{6} + 19853371612542086 T^{8} - 2648556220200 p^{4} T^{10} + 289539196 p^{8} T^{12} - 19480 p^{12} T^{14} + p^{16} T^{16} \) |
| 83 | \( 1 + 980 T^{2} + 49404756 T^{4} + 355567312300 T^{6} + 2834941205538326 T^{8} + 355567312300 p^{4} T^{10} + 49404756 p^{8} T^{12} + 980 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 - 110 T + 20657 T^{2} - 1930370 T^{3} + 245768268 T^{4} - 1930370 p^{2} T^{5} + 20657 p^{4} T^{6} - 110 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 4 T + 30820 T^{2} - 154428 T^{3} + 410716598 T^{4} - 154428 p^{2} T^{5} + 30820 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.91236070077916206465907960992, −3.89708583458309885828848475042, −3.77439557335367413989705717244, −3.67545400396807706891286091046, −3.35674155853489851832795326786, −3.21187951710206860130024400685, −3.11314269492470116930415130141, −3.04736158736259976688346958667, −3.02163214914010747775416127523, −2.76791868166167802884529466981, −2.62428309111375672287975666144, −2.59424848498224001835739408204, −2.37669566422374290195358269679, −2.11101688698659645658216281812, −2.10995734993204871729694823716, −1.98811110003249855022817966545, −1.79153770688133166670528255821, −1.69269224796852592522336116388, −1.39285295731850412234256445834, −1.16185356376559665454435946158, −1.15952192175323513860038068970, −0.844589032380140134995607767404, −0.50832468429743735143525845271, −0.35940251098028265508182805948, −0.24084848063455514285839829092,
0.24084848063455514285839829092, 0.35940251098028265508182805948, 0.50832468429743735143525845271, 0.844589032380140134995607767404, 1.15952192175323513860038068970, 1.16185356376559665454435946158, 1.39285295731850412234256445834, 1.69269224796852592522336116388, 1.79153770688133166670528255821, 1.98811110003249855022817966545, 2.10995734993204871729694823716, 2.11101688698659645658216281812, 2.37669566422374290195358269679, 2.59424848498224001835739408204, 2.62428309111375672287975666144, 2.76791868166167802884529466981, 3.02163214914010747775416127523, 3.04736158736259976688346958667, 3.11314269492470116930415130141, 3.21187951710206860130024400685, 3.35674155853489851832795326786, 3.67545400396807706891286091046, 3.77439557335367413989705717244, 3.89708583458309885828848475042, 3.91236070077916206465907960992
Plot not available for L-functions of degree greater than 10.
