| L(s) = 1 | + 1.73·3-s − 5-s − 1.25·7-s + 0.00617·9-s + 2.50·13-s − 1.73·15-s + 3.60·17-s − 7.05·19-s − 2.17·21-s + 2.20·23-s + 25-s − 5.19·27-s − 2.69·29-s + 2.83·31-s + 1.25·35-s − 8.07·37-s + 4.34·39-s + 10.3·41-s + 6.88·43-s − 0.00617·45-s − 1.59·47-s − 5.43·49-s + 6.24·51-s + 1.77·53-s − 12.2·57-s + 4.36·59-s + 8.33·61-s + ⋯ |
| L(s) = 1 | + 1.00·3-s − 0.447·5-s − 0.473·7-s + 0.00205·9-s + 0.695·13-s − 0.447·15-s + 0.874·17-s − 1.61·19-s − 0.473·21-s + 0.460·23-s + 0.200·25-s − 0.998·27-s − 0.501·29-s + 0.508·31-s + 0.211·35-s − 1.32·37-s + 0.696·39-s + 1.61·41-s + 1.05·43-s − 0.000921·45-s − 0.232·47-s − 0.775·49-s + 0.875·51-s + 0.243·53-s − 1.62·57-s + 0.568·59-s + 1.06·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.288038170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.288038170\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 1.73T + 3T^{2} \) |
| 7 | \( 1 + 1.25T + 7T^{2} \) |
| 13 | \( 1 - 2.50T + 13T^{2} \) |
| 17 | \( 1 - 3.60T + 17T^{2} \) |
| 19 | \( 1 + 7.05T + 19T^{2} \) |
| 23 | \( 1 - 2.20T + 23T^{2} \) |
| 29 | \( 1 + 2.69T + 29T^{2} \) |
| 31 | \( 1 - 2.83T + 31T^{2} \) |
| 37 | \( 1 + 8.07T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 6.88T + 43T^{2} \) |
| 47 | \( 1 + 1.59T + 47T^{2} \) |
| 53 | \( 1 - 1.77T + 53T^{2} \) |
| 59 | \( 1 - 4.36T + 59T^{2} \) |
| 61 | \( 1 - 8.33T + 61T^{2} \) |
| 67 | \( 1 + 5.43T + 67T^{2} \) |
| 71 | \( 1 - 4.63T + 71T^{2} \) |
| 73 | \( 1 - 2.03T + 73T^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 + 5.62T + 83T^{2} \) |
| 89 | \( 1 + 4.72T + 89T^{2} \) |
| 97 | \( 1 - 19.0T + 97T^{2} \) |
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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
−7.81128145636715914309330153188, −7.14248004127109154479107772943, −6.32827157467073757269446067902, −5.76576628885024708042994639865, −4.79418191862563690750097754748, −3.87281414114363094492645057075, −3.50443398430394104743470959979, −2.68354075089112795594847405675, −1.92683926558309815776409552994, −0.66820516707056679494995137469,
0.66820516707056679494995137469, 1.92683926558309815776409552994, 2.68354075089112795594847405675, 3.50443398430394104743470959979, 3.87281414114363094492645057075, 4.79418191862563690750097754748, 5.76576628885024708042994639865, 6.32827157467073757269446067902, 7.14248004127109154479107772943, 7.81128145636715914309330153188
