| L(s) = 1 | + 2-s + 2.30·3-s + 4-s − 1.83·5-s + 2.30·6-s + 8-s + 2.29·9-s − 1.83·10-s − 4.13·11-s + 2.30·12-s − 5.59·13-s − 4.22·15-s + 16-s − 0.628·17-s + 2.29·18-s − 0.404·19-s − 1.83·20-s − 4.13·22-s − 1.92·23-s + 2.30·24-s − 1.62·25-s − 5.59·26-s − 1.62·27-s + 3.43·29-s − 4.22·30-s − 0.705·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.32·3-s + 0.5·4-s − 0.821·5-s + 0.939·6-s + 0.353·8-s + 0.764·9-s − 0.580·10-s − 1.24·11-s + 0.664·12-s − 1.55·13-s − 1.09·15-s + 0.250·16-s − 0.152·17-s + 0.540·18-s − 0.0927·19-s − 0.410·20-s − 0.882·22-s − 0.401·23-s + 0.469·24-s − 0.325·25-s − 1.09·26-s − 0.312·27-s + 0.637·29-s − 0.771·30-s − 0.126·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 2.30T + 3T^{2} \) |
| 5 | \( 1 + 1.83T + 5T^{2} \) |
| 11 | \( 1 + 4.13T + 11T^{2} \) |
| 13 | \( 1 + 5.59T + 13T^{2} \) |
| 17 | \( 1 + 0.628T + 17T^{2} \) |
| 19 | \( 1 + 0.404T + 19T^{2} \) |
| 23 | \( 1 + 1.92T + 23T^{2} \) |
| 29 | \( 1 - 3.43T + 29T^{2} \) |
| 31 | \( 1 + 0.705T + 31T^{2} \) |
| 37 | \( 1 - 2.60T + 37T^{2} \) |
| 43 | \( 1 - 2.37T + 43T^{2} \) |
| 47 | \( 1 + 4.67T + 47T^{2} \) |
| 53 | \( 1 + 9.70T + 53T^{2} \) |
| 59 | \( 1 + 2.38T + 59T^{2} \) |
| 61 | \( 1 + 0.491T + 61T^{2} \) |
| 67 | \( 1 + 7.06T + 67T^{2} \) |
| 71 | \( 1 - 9.89T + 71T^{2} \) |
| 73 | \( 1 + 7.19T + 73T^{2} \) |
| 79 | \( 1 - 7.63T + 79T^{2} \) |
| 83 | \( 1 + 6.21T + 83T^{2} \) |
| 89 | \( 1 + 8.70T + 89T^{2} \) |
| 97 | \( 1 + 2.81T + 97T^{2} \) |
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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
−7.81355134768749685261184758135, −7.68413492424305915530077099962, −6.81029487434737572607090865820, −5.69964837897626235392383163332, −4.81437210724293323826717393740, −4.23881712565802281591183138497, −3.26063138750942650815820701551, −2.71245273900437060553579855274, −1.98247498361068665404075164520, 0,
1.98247498361068665404075164520, 2.71245273900437060553579855274, 3.26063138750942650815820701551, 4.23881712565802281591183138497, 4.81437210724293323826717393740, 5.69964837897626235392383163332, 6.81029487434737572607090865820, 7.68413492424305915530077099962, 7.81355134768749685261184758135
