| L(s) = 1 | + 2.30·2-s + 3.30·4-s − 2.30·5-s − 0.302·7-s + 3.00·8-s − 5.30·10-s − 3·11-s − 3.30·13-s − 0.697·14-s + 0.302·16-s + 5.90·19-s − 7.60·20-s − 6.90·22-s − 2.30·23-s + 0.302·25-s − 7.60·26-s − 1.00·28-s − 9.90·29-s + 3.60·31-s − 5.30·32-s + 0.697·35-s + 0.605·37-s + 13.6·38-s − 6.90·40-s − 6·41-s − 2.39·43-s − 9.90·44-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 1.65·4-s − 1.02·5-s − 0.114·7-s + 1.06·8-s − 1.67·10-s − 0.904·11-s − 0.916·13-s − 0.186·14-s + 0.0756·16-s + 1.35·19-s − 1.70·20-s − 1.47·22-s − 0.480·23-s + 0.0605·25-s − 1.49·26-s − 0.188·28-s − 1.83·29-s + 0.647·31-s − 0.937·32-s + 0.117·35-s + 0.0995·37-s + 2.20·38-s − 1.09·40-s − 0.937·41-s − 0.365·43-s − 1.49·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 5 | \( 1 + 2.30T + 5T^{2} \) |
| 7 | \( 1 + 0.302T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 3.30T + 13T^{2} \) |
| 19 | \( 1 - 5.90T + 19T^{2} \) |
| 23 | \( 1 + 2.30T + 23T^{2} \) |
| 29 | \( 1 + 9.90T + 29T^{2} \) |
| 31 | \( 1 - 3.60T + 31T^{2} \) |
| 37 | \( 1 - 0.605T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 2.39T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 + 2.09T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 8.81T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 3.21T + 71T^{2} \) |
| 73 | \( 1 - 0.394T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 + 2.51T + 83T^{2} \) |
| 89 | \( 1 + 3.21T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.076959310483615455380026989362, −7.58307332901373472304837379994, −6.90699841512484401428617196023, −5.90102604941539825845694409105, −5.14610034585046777705406875831, −4.62005426137955655927307288412, −3.61215839397958044673276825494, −3.12369683443297215954463367628, −2.04473626427702883805702296792, 0,
2.04473626427702883805702296792, 3.12369683443297215954463367628, 3.61215839397958044673276825494, 4.62005426137955655927307288412, 5.14610034585046777705406875831, 5.90102604941539825845694409105, 6.90699841512484401428617196023, 7.58307332901373472304837379994, 8.076959310483615455380026989362
