| L(s) = 1 | + 0.948·2-s − 0.163·3-s − 7.10·4-s − 5.36·5-s − 0.154·6-s − 7·7-s − 14.3·8-s − 26.9·9-s − 5.09·10-s + 11·11-s + 1.16·12-s − 42.9·13-s − 6.64·14-s + 0.877·15-s + 43.2·16-s − 60.8·17-s − 25.5·18-s + 140.·19-s + 38.1·20-s + 1.14·21-s + 10.4·22-s − 91.3·23-s + 2.34·24-s − 96.1·25-s − 40.7·26-s + 8.81·27-s + 49.7·28-s + ⋯ |
| L(s) = 1 | + 0.335·2-s − 0.0314·3-s − 0.887·4-s − 0.480·5-s − 0.0105·6-s − 0.377·7-s − 0.633·8-s − 0.999·9-s − 0.161·10-s + 0.301·11-s + 0.0279·12-s − 0.916·13-s − 0.126·14-s + 0.0150·15-s + 0.675·16-s − 0.868·17-s − 0.335·18-s + 1.69·19-s + 0.426·20-s + 0.0118·21-s + 0.101·22-s − 0.828·23-s + 0.0199·24-s − 0.769·25-s − 0.307·26-s + 0.0628·27-s + 0.335·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 7T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 0.948T + 8T^{2} \) |
| 3 | \( 1 + 0.163T + 27T^{2} \) |
| 5 | \( 1 + 5.36T + 125T^{2} \) |
| 13 | \( 1 + 42.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 60.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 140.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 91.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 260.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 259.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 359.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 320.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 92.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 67.4T + 1.03e5T^{2} \) |
| 53 | \( 1 + 246.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 475.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 799.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 725.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 544.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 580.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 402.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.10e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.25e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 9.99e2T + 9.12e5T^{2} \) |
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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
−13.62046537126141447944414770767, −12.31064696667285712597482689090, −11.55420757730571156288872017611, −9.889853972950125312718133112692, −8.928873679924078421847383066045, −7.68448651941798088506216457941, −5.97240785298497489268845710408, −4.65625976084825097502996763721, −3.18550746492473304821872059558, 0,
3.18550746492473304821872059558, 4.65625976084825097502996763721, 5.97240785298497489268845710408, 7.68448651941798088506216457941, 8.928873679924078421847383066045, 9.889853972950125312718133112692, 11.55420757730571156288872017611, 12.31064696667285712597482689090, 13.62046537126141447944414770767
