| L(s) = 1 | + 2-s + 2.30·3-s + 4-s − 1.30·5-s + 2.30·6-s − 7-s + 8-s + 2.30·9-s − 1.30·10-s + 3·11-s + 2.30·12-s + 5·13-s − 14-s − 3·15-s + 16-s − 5.60·17-s + 2.30·18-s + 2·19-s − 1.30·20-s − 2.30·21-s + 3·22-s + 8.21·23-s + 2.30·24-s − 3.30·25-s + 5·26-s − 1.60·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.32·3-s + 0.5·4-s − 0.582·5-s + 0.940·6-s − 0.377·7-s + 0.353·8-s + 0.767·9-s − 0.411·10-s + 0.904·11-s + 0.664·12-s + 1.38·13-s − 0.267·14-s − 0.774·15-s + 0.250·16-s − 1.35·17-s + 0.542·18-s + 0.458·19-s − 0.291·20-s − 0.502·21-s + 0.639·22-s + 1.71·23-s + 0.470·24-s − 0.660·25-s + 0.980·26-s − 0.308·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.041515424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.041515424\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 269 | \( 1 + T \) |
| good | 3 | \( 1 - 2.30T + 3T^{2} \) |
| 5 | \( 1 + 1.30T + 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 + 5.60T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 8.21T + 23T^{2} \) |
| 29 | \( 1 - 4.69T + 29T^{2} \) |
| 31 | \( 1 + 9.60T + 31T^{2} \) |
| 37 | \( 1 + 3.60T + 37T^{2} \) |
| 41 | \( 1 + 8.60T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 + 9.90T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 6.51T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - 16.2T + 67T^{2} \) |
| 71 | \( 1 - 7.81T + 71T^{2} \) |
| 73 | \( 1 + 3.60T + 73T^{2} \) |
| 79 | \( 1 + 2.69T + 79T^{2} \) |
| 83 | \( 1 - 8.60T + 83T^{2} \) |
| 89 | \( 1 + 7.30T + 89T^{2} \) |
| 97 | \( 1 - 4.60T + 97T^{2} \) |
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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
−11.14746901950943647195748594240, −9.776204173864185958574316016086, −8.801734598099893482141580861158, −8.370356655563098782338215071683, −7.08655305238369845374278922641, −6.45769168146058318023883440281, −4.94760952852127934613324027835, −3.62739899889701253609734948047, −3.37919496802167730530231627105, −1.79846874068588125902600475991,
1.79846874068588125902600475991, 3.37919496802167730530231627105, 3.62739899889701253609734948047, 4.94760952852127934613324027835, 6.45769168146058318023883440281, 7.08655305238369845374278922641, 8.370356655563098782338215071683, 8.801734598099893482141580861158, 9.776204173864185958574316016086, 11.14746901950943647195748594240
