| L(s) = 1 | + 2·4-s + 4·7-s − 12·13-s + 16-s − 16·25-s + 8·28-s + 12·31-s − 4·37-s − 28·43-s + 18·49-s − 24·52-s − 12·61-s − 2·64-s − 20·67-s − 72·73-s − 20·79-s − 48·91-s − 12·97-s − 32·100-s + 20·109-s + 4·112-s + 16·121-s + 24·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 4-s + 1.51·7-s − 3.32·13-s + 1/4·16-s − 3.19·25-s + 1.51·28-s + 2.15·31-s − 0.657·37-s − 4.26·43-s + 18/7·49-s − 3.32·52-s − 1.53·61-s − 1/4·64-s − 2.44·67-s − 8.42·73-s − 2.25·79-s − 5.03·91-s − 1.21·97-s − 3.19·100-s + 1.91·109-s + 0.377·112-s + 1.45·121-s + 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.489708902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.489708902\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 5 | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 6 T + 35 T^{2} + 138 T^{3} + 516 T^{4} + 138 p T^{5} + 35 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 28 T^{2} + 495 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 14 T^{2} - 165 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( 1 + 8 T^{2} + 958 T^{4} - 20608 T^{6} + 38131 T^{8} - 20608 p^{2} T^{10} + 958 p^{4} T^{12} + 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 6 T + 23 T^{2} - 66 T^{3} - 468 T^{4} - 66 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 2 T - 53 T^{2} - 34 T^{3} + 1732 T^{4} - 34 p T^{5} - 53 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 76 T^{2} + 4095 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 7 T + 6 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( 1 + 40 T^{2} - 626 T^{4} - 87680 T^{6} - 3459005 T^{8} - 87680 p^{2} T^{10} - 626 p^{4} T^{12} + 40 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 - 4 T^{2} + 4762 T^{4} + 41456 T^{6} + 14589091 T^{8} + 41456 p^{2} T^{10} + 4762 p^{4} T^{12} - 4 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 112 T^{2} + 9063 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 6 T + 131 T^{2} + 714 T^{3} + 11172 T^{4} + 714 p T^{5} + 131 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 10 T + 13 T^{2} - 470 T^{3} - 3620 T^{4} - 470 p T^{5} + 13 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 36 T + 662 T^{2} + 8280 T^{3} + 79107 T^{4} + 8280 p T^{5} + 662 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 10 T - 65 T^{2} + 70 T^{3} + 13084 T^{4} + 70 p T^{5} - 65 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 68 T^{2} + 58 T^{4} + 626416 T^{6} - 58464701 T^{8} + 626416 p^{2} T^{10} + 58 p^{4} T^{12} - 68 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 - 32 T^{2} + 8254 T^{4} + 738304 T^{6} - 18580445 T^{8} + 738304 p^{2} T^{10} + 8254 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 + 6 T + 185 T^{2} + 1038 T^{3} + 21684 T^{4} + 1038 p T^{5} + 185 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.37894277570026584570274558987, −4.22003702026723668411839997607, −4.00438830800858746094855578152, −3.91306851935200457229722900909, −3.59214993218178652562205054395, −3.33363051843523330026535384511, −3.32341555593014793211934535159, −3.20586065903968593582381807646, −3.12514507161898401816594520101, −3.03380757797395609866656792771, −2.83073500613375217699295336433, −2.69711651946668094716947870572, −2.46127306876606392418997200414, −2.44302344720756055155918673236, −2.31635502052923610000212579978, −1.95768773396783226594213700264, −1.82692531123717948946328657958, −1.75284767361655819494280799994, −1.73291390937758125722052482363, −1.52324145425655715529899315153, −1.39158279958045387003945535096, −1.24410908931344260854396533071, −0.44155875204908229132988082584, −0.41643311164452117719419900668, −0.34058452653445494379636304857,
0.34058452653445494379636304857, 0.41643311164452117719419900668, 0.44155875204908229132988082584, 1.24410908931344260854396533071, 1.39158279958045387003945535096, 1.52324145425655715529899315153, 1.73291390937758125722052482363, 1.75284767361655819494280799994, 1.82692531123717948946328657958, 1.95768773396783226594213700264, 2.31635502052923610000212579978, 2.44302344720756055155918673236, 2.46127306876606392418997200414, 2.69711651946668094716947870572, 2.83073500613375217699295336433, 3.03380757797395609866656792771, 3.12514507161898401816594520101, 3.20586065903968593582381807646, 3.32341555593014793211934535159, 3.33363051843523330026535384511, 3.59214993218178652562205054395, 3.91306851935200457229722900909, 4.00438830800858746094855578152, 4.22003702026723668411839997607, 4.37894277570026584570274558987
Plot not available for L-functions of degree greater than 10.
