| L(s) = 1 | + 32·2-s + 1.02e3·4-s + 3.12e3·5-s + 2.55e4·7-s + 3.27e4·8-s + 1.00e5·10-s − 7.69e5·11-s − 9.18e5·13-s + 8.18e5·14-s + 1.04e6·16-s − 1.03e7·17-s − 5.52e6·19-s + 3.20e6·20-s − 2.46e7·22-s + 3.99e7·23-s + 9.76e6·25-s − 2.94e7·26-s + 2.61e7·28-s + 1.52e7·29-s − 2.41e8·31-s + 3.35e7·32-s − 3.30e8·34-s + 7.99e7·35-s − 2.57e7·37-s − 1.76e8·38-s + 1.02e8·40-s + 1.21e9·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.575·7-s + 0.353·8-s + 0.316·10-s − 1.43·11-s − 0.686·13-s + 0.406·14-s + 1/4·16-s − 1.76·17-s − 0.511·19-s + 0.223·20-s − 1.01·22-s + 1.29·23-s + 1/5·25-s − 0.485·26-s + 0.287·28-s + 0.138·29-s − 1.51·31-s + 0.176·32-s − 1.24·34-s + 0.257·35-s − 0.0610·37-s − 0.361·38-s + 0.158·40-s + 1.64·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{5} T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p^{5} T \) |
| good | 7 | \( 1 - 25574 T + p^{11} T^{2} \) |
| 11 | \( 1 + 769152 T + p^{11} T^{2} \) |
| 13 | \( 1 + 918982 T + p^{11} T^{2} \) |
| 17 | \( 1 + 10312794 T + p^{11} T^{2} \) |
| 19 | \( 1 + 5521660 T + p^{11} T^{2} \) |
| 23 | \( 1 - 39973422 T + p^{11} T^{2} \) |
| 29 | \( 1 - 15269010 T + p^{11} T^{2} \) |
| 31 | \( 1 + 241583788 T + p^{11} T^{2} \) |
| 37 | \( 1 + 25751446 T + p^{11} T^{2} \) |
| 41 | \( 1 - 1217700138 T + p^{11} T^{2} \) |
| 43 | \( 1 + 683436262 T + p^{11} T^{2} \) |
| 47 | \( 1 + 1537395294 T + p^{11} T^{2} \) |
| 53 | \( 1 + 3572891298 T + p^{11} T^{2} \) |
| 59 | \( 1 - 1069039020 T + p^{11} T^{2} \) |
| 61 | \( 1 + 2091535078 T + p^{11} T^{2} \) |
| 67 | \( 1 + 1462369186 T + p^{11} T^{2} \) |
| 71 | \( 1 + 9660178332 T + p^{11} T^{2} \) |
| 73 | \( 1 + 5603447662 T + p^{11} T^{2} \) |
| 79 | \( 1 - 5026936280 T + p^{11} T^{2} \) |
| 83 | \( 1 - 38405955462 T + p^{11} T^{2} \) |
| 89 | \( 1 + 35558583210 T + p^{11} T^{2} \) |
| 97 | \( 1 - 10572232514 T + p^{11} T^{2} \) |
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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
−11.24346578161624575367077648324, −10.61556959607151458089911417389, −9.158065252457685323528125502356, −7.82899992821323037348039595224, −6.68724184267817470196097084822, −5.31369142936042861159062934829, −4.55772110326584347978498203325, −2.80854216475562840693400822621, −1.87825577844726920611529392427, 0,
1.87825577844726920611529392427, 2.80854216475562840693400822621, 4.55772110326584347978498203325, 5.31369142936042861159062934829, 6.68724184267817470196097084822, 7.82899992821323037348039595224, 9.158065252457685323528125502356, 10.61556959607151458089911417389, 11.24346578161624575367077648324
