| L(s) = 1 | + 1.58·3-s − 1.03·5-s − 0.473·9-s + 0.432·11-s − 1.93·13-s − 1.64·15-s + 5.83·17-s + 3.09·19-s − 23-s − 3.92·25-s − 5.52·27-s − 7.12·29-s + 5.54·31-s + 0.688·33-s − 9.75·37-s − 3.06·39-s − 7.95·41-s + 6.55·43-s + 0.490·45-s + 6.54·47-s + 9.27·51-s + 14.2·53-s − 0.448·55-s + 4.92·57-s − 7.77·59-s − 1.28·61-s + 2.00·65-s + ⋯ |
| L(s) = 1 | + 0.917·3-s − 0.463·5-s − 0.157·9-s + 0.130·11-s − 0.535·13-s − 0.425·15-s + 1.41·17-s + 0.710·19-s − 0.208·23-s − 0.784·25-s − 1.06·27-s − 1.32·29-s + 0.995·31-s + 0.119·33-s − 1.60·37-s − 0.491·39-s − 1.24·41-s + 0.999·43-s + 0.0731·45-s + 0.955·47-s + 1.29·51-s + 1.96·53-s − 0.0605·55-s + 0.651·57-s − 1.01·59-s − 0.164·61-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 1.58T + 3T^{2} \) |
| 5 | \( 1 + 1.03T + 5T^{2} \) |
| 11 | \( 1 - 0.432T + 11T^{2} \) |
| 13 | \( 1 + 1.93T + 13T^{2} \) |
| 17 | \( 1 - 5.83T + 17T^{2} \) |
| 19 | \( 1 - 3.09T + 19T^{2} \) |
| 29 | \( 1 + 7.12T + 29T^{2} \) |
| 31 | \( 1 - 5.54T + 31T^{2} \) |
| 37 | \( 1 + 9.75T + 37T^{2} \) |
| 41 | \( 1 + 7.95T + 41T^{2} \) |
| 43 | \( 1 - 6.55T + 43T^{2} \) |
| 47 | \( 1 - 6.54T + 47T^{2} \) |
| 53 | \( 1 - 14.2T + 53T^{2} \) |
| 59 | \( 1 + 7.77T + 59T^{2} \) |
| 61 | \( 1 + 1.28T + 61T^{2} \) |
| 67 | \( 1 + 3.35T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 8.75T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 - 6.39T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
| 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
−7.59322680736431507049476286339, −7.02299849957492800948953414272, −5.79601525746000841926320056744, −5.49720586009805120389355339541, −4.42579024791443684348899485844, −3.63353778876326242948033372882, −3.17114642839129585089291516339, −2.32130648265039878354714279021, −1.36457301973503209670781836532, 0,
1.36457301973503209670781836532, 2.32130648265039878354714279021, 3.17114642839129585089291516339, 3.63353778876326242948033372882, 4.42579024791443684348899485844, 5.49720586009805120389355339541, 5.79601525746000841926320056744, 7.02299849957492800948953414272, 7.59322680736431507049476286339
