| L(s) = 1 | + 2-s − 1.32·3-s + 4-s − 4.14·5-s − 1.32·6-s − 0.0466·7-s + 8-s − 1.25·9-s − 4.14·10-s + 1.23·11-s − 1.32·12-s − 1.17·13-s − 0.0466·14-s + 5.48·15-s + 16-s + 5.43·17-s − 1.25·18-s + 0.299·19-s − 4.14·20-s + 0.0617·21-s + 1.23·22-s + 1.80·23-s − 1.32·24-s + 12.1·25-s − 1.17·26-s + 5.62·27-s − 0.0466·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.763·3-s + 0.5·4-s − 1.85·5-s − 0.540·6-s − 0.0176·7-s + 0.353·8-s − 0.416·9-s − 1.31·10-s + 0.371·11-s − 0.381·12-s − 0.324·13-s − 0.0124·14-s + 1.41·15-s + 0.250·16-s + 1.31·17-s − 0.294·18-s + 0.0687·19-s − 0.926·20-s + 0.0134·21-s + 0.262·22-s + 0.376·23-s − 0.270·24-s + 2.43·25-s − 0.229·26-s + 1.08·27-s − 0.00882·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 + 1.32T + 3T^{2} \) |
| 5 | \( 1 + 4.14T + 5T^{2} \) |
| 7 | \( 1 + 0.0466T + 7T^{2} \) |
| 11 | \( 1 - 1.23T + 11T^{2} \) |
| 13 | \( 1 + 1.17T + 13T^{2} \) |
| 17 | \( 1 - 5.43T + 17T^{2} \) |
| 19 | \( 1 - 0.299T + 19T^{2} \) |
| 23 | \( 1 - 1.80T + 23T^{2} \) |
| 29 | \( 1 - 5.16T + 29T^{2} \) |
| 31 | \( 1 + 2.12T + 31T^{2} \) |
| 37 | \( 1 - 4.58T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 4.77T + 43T^{2} \) |
| 47 | \( 1 + 4.85T + 47T^{2} \) |
| 53 | \( 1 + 0.552T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 + 1.23T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 - 0.575T + 71T^{2} \) |
| 73 | \( 1 + 4.13T + 73T^{2} \) |
| 79 | \( 1 + 1.84T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + 6.58T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−7.964096614557889599454292488278, −7.26116912635096881657475477604, −6.59214877628085864824260692937, −5.76890737862470129004173500955, −4.89624253064326180154118146872, −4.43917614591563635151597283876, −3.38763583278509486106451545449, −3.00392924705072213027496787175, −1.19173469229719508878000723335, 0,
1.19173469229719508878000723335, 3.00392924705072213027496787175, 3.38763583278509486106451545449, 4.43917614591563635151597283876, 4.89624253064326180154118146872, 5.76890737862470129004173500955, 6.59214877628085864824260692937, 7.26116912635096881657475477604, 7.964096614557889599454292488278
