| L(s) = 1 | + 2-s − 3.19·3-s + 4-s − 0.357·5-s − 3.19·6-s + 4.43·7-s + 8-s + 7.22·9-s − 0.357·10-s + 11-s − 3.19·12-s − 0.308·13-s + 4.43·14-s + 1.14·15-s + 16-s + 2.41·17-s + 7.22·18-s − 0.357·20-s − 14.1·21-s + 22-s + 1.48·23-s − 3.19·24-s − 4.87·25-s − 0.308·26-s − 13.5·27-s + 4.43·28-s + 3.81·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.84·3-s + 0.5·4-s − 0.159·5-s − 1.30·6-s + 1.67·7-s + 0.353·8-s + 2.40·9-s − 0.112·10-s + 0.301·11-s − 0.923·12-s − 0.0855·13-s + 1.18·14-s + 0.294·15-s + 0.250·16-s + 0.585·17-s + 1.70·18-s − 0.0798·20-s − 3.09·21-s + 0.213·22-s + 0.309·23-s − 0.652·24-s − 0.974·25-s − 0.0605·26-s − 2.60·27-s + 0.838·28-s + 0.708·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.472226578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.472226578\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 3.19T + 3T^{2} \) |
| 5 | \( 1 + 0.357T + 5T^{2} \) |
| 7 | \( 1 - 4.43T + 7T^{2} \) |
| 13 | \( 1 + 0.308T + 13T^{2} \) |
| 17 | \( 1 - 2.41T + 17T^{2} \) |
| 23 | \( 1 - 1.48T + 23T^{2} \) |
| 29 | \( 1 - 3.81T + 29T^{2} \) |
| 31 | \( 1 - 9.99T + 31T^{2} \) |
| 37 | \( 1 - 6.92T + 37T^{2} \) |
| 41 | \( 1 - 6.29T + 41T^{2} \) |
| 43 | \( 1 - 3.24T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 9.44T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 3.86T + 61T^{2} \) |
| 67 | \( 1 - 1.56T + 67T^{2} \) |
| 71 | \( 1 + 5.02T + 71T^{2} \) |
| 73 | \( 1 - 3.27T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 - 0.214T + 83T^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
| 97 | \( 1 + 6.87T + 97T^{2} \) |
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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
−7.70765065233152385244017041765, −6.97878636299900059008095323585, −6.12931266789616175660699900108, −5.76701836901238228646264845793, −4.99205585844099460995512862084, −4.48856658487171253324835360418, −4.07950159179704823154657771245, −2.62265491933542058935454446081, −1.48018880677349268512691998846, −0.872856044998411087452272390753,
0.872856044998411087452272390753, 1.48018880677349268512691998846, 2.62265491933542058935454446081, 4.07950159179704823154657771245, 4.48856658487171253324835360418, 4.99205585844099460995512862084, 5.76701836901238228646264845793, 6.12931266789616175660699900108, 6.97878636299900059008095323585, 7.70765065233152385244017041765
