| L(s) = 1 | + 27·3-s − 160·5-s + 974·7-s + 729·9-s − 3.95e3·11-s − 574·13-s − 4.32e3·15-s − 8.47e3·17-s + 5.33e4·19-s + 2.62e4·21-s − 9.84e4·23-s − 5.25e4·25-s + 1.96e4·27-s − 5.10e4·29-s + 2.05e5·31-s − 1.06e5·33-s − 1.55e5·35-s − 2.55e5·37-s − 1.54e4·39-s − 6.65e5·41-s + 3.96e5·43-s − 1.16e5·45-s + 5.49e5·47-s + 1.25e5·49-s − 2.28e5·51-s + 7.20e5·53-s + 6.32e5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.572·5-s + 1.07·7-s + 1/3·9-s − 0.896·11-s − 0.0724·13-s − 0.330·15-s − 0.418·17-s + 1.78·19-s + 0.619·21-s − 1.68·23-s − 0.672·25-s + 0.192·27-s − 0.388·29-s + 1.23·31-s − 0.517·33-s − 0.614·35-s − 0.829·37-s − 0.0418·39-s − 1.50·41-s + 0.761·43-s − 0.190·45-s + 0.771·47-s + 0.151·49-s − 0.241·51-s + 0.664·53-s + 0.512·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{3} T \) |
| good | 5 | \( 1 + 32 p T + p^{7} T^{2} \) |
| 7 | \( 1 - 974 T + p^{7} T^{2} \) |
| 11 | \( 1 + 3956 T + p^{7} T^{2} \) |
| 13 | \( 1 + 574 T + p^{7} T^{2} \) |
| 17 | \( 1 + 8474 T + p^{7} T^{2} \) |
| 19 | \( 1 - 53312 T + p^{7} T^{2} \) |
| 23 | \( 1 + 98468 T + p^{7} T^{2} \) |
| 29 | \( 1 + 51060 T + p^{7} T^{2} \) |
| 31 | \( 1 - 205014 T + p^{7} T^{2} \) |
| 37 | \( 1 + 255674 T + p^{7} T^{2} \) |
| 41 | \( 1 + 665394 T + p^{7} T^{2} \) |
| 43 | \( 1 - 396840 T + p^{7} T^{2} \) |
| 47 | \( 1 - 549388 T + p^{7} T^{2} \) |
| 53 | \( 1 - 720060 T + p^{7} T^{2} \) |
| 59 | \( 1 + 1043956 T + p^{7} T^{2} \) |
| 61 | \( 1 - 2055734 T + p^{7} T^{2} \) |
| 67 | \( 1 + 2092652 T + p^{7} T^{2} \) |
| 71 | \( 1 + 2723868 T + p^{7} T^{2} \) |
| 73 | \( 1 + 5190010 T + p^{7} T^{2} \) |
| 79 | \( 1 - 3647110 T + p^{7} T^{2} \) |
| 83 | \( 1 + 2101956 T + p^{7} T^{2} \) |
| 89 | \( 1 - 522514 T + p^{7} T^{2} \) |
| 97 | \( 1 + 1361990 T + p^{7} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.736583726452071941552442616984, −8.513735502969826976793620535040, −7.894284270939187551896261150738, −7.26261213682747463621524853836, −5.69724719660895288363671905699, −4.72589087499101181220972371828, −3.72165965746318703148367597642, −2.52914538555211660513665287681, −1.43668505862262532393942380411, 0,
1.43668505862262532393942380411, 2.52914538555211660513665287681, 3.72165965746318703148367597642, 4.72589087499101181220972371828, 5.69724719660895288363671905699, 7.26261213682747463621524853836, 7.894284270939187551896261150738, 8.513735502969826976793620535040, 9.736583726452071941552442616984
