| L(s) = 1 | − 1.32·2-s − 2.14·3-s − 0.236·4-s + 2.85·6-s + 3.47·7-s + 2.96·8-s + 1.61·9-s + 2·11-s + 0.507·12-s + 2.65·13-s − 4.61·14-s − 3.47·16-s − 4.29·17-s − 2.14·18-s + 7.23·19-s − 7.47·21-s − 2.65·22-s + 0.820·23-s − 6.38·24-s − 3.52·26-s + 2.96·27-s − 0.820·28-s + 0.854·29-s + 2·31-s − 1.32·32-s − 4.29·33-s + 5.70·34-s + ⋯ |
| L(s) = 1 | − 0.939·2-s − 1.24·3-s − 0.118·4-s + 1.16·6-s + 1.31·7-s + 1.04·8-s + 0.539·9-s + 0.603·11-s + 0.146·12-s + 0.736·13-s − 1.23·14-s − 0.868·16-s − 1.04·17-s − 0.506·18-s + 1.66·19-s − 1.63·21-s − 0.566·22-s + 0.171·23-s − 1.30·24-s − 0.691·26-s + 0.571·27-s − 0.155·28-s + 0.158·29-s + 0.359·31-s − 0.234·32-s − 0.748·33-s + 0.978·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4940857591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4940857591\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 1.32T + 2T^{2} \) |
| 3 | \( 1 + 2.14T + 3T^{2} \) |
| 7 | \( 1 - 3.47T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2.65T + 13T^{2} \) |
| 17 | \( 1 + 4.29T + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 23 | \( 1 - 0.820T + 23T^{2} \) |
| 29 | \( 1 - 0.854T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 1.64T + 37T^{2} \) |
| 41 | \( 1 + 6.09T + 41T^{2} \) |
| 43 | \( 1 - 3.79T + 43T^{2} \) |
| 47 | \( 1 - 0.507T + 47T^{2} \) |
| 53 | \( 1 - 8.59T + 53T^{2} \) |
| 59 | \( 1 + 4.47T + 59T^{2} \) |
| 61 | \( 1 - 5.09T + 61T^{2} \) |
| 67 | \( 1 + 4.29T + 67T^{2} \) |
| 71 | \( 1 - 8.18T + 71T^{2} \) |
| 73 | \( 1 + 16.5T + 73T^{2} \) |
| 79 | \( 1 - 2.76T + 79T^{2} \) |
| 83 | \( 1 + 5.11T + 83T^{2} \) |
| 89 | \( 1 + 8.61T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
| 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
−13.41615861849473507220117410601, −11.82975374775973850453887254620, −11.27893146477352786074014132853, −10.43948827731523487968532120433, −9.139624370237974631505965024215, −8.184815295102623510670754014241, −6.94738546703140403774503923250, −5.47438037871621123375050606667, −4.43816783911439271747278038055, −1.21743884198223694890758351404,
1.21743884198223694890758351404, 4.43816783911439271747278038055, 5.47438037871621123375050606667, 6.94738546703140403774503923250, 8.184815295102623510670754014241, 9.139624370237974631505965024215, 10.43948827731523487968532120433, 11.27893146477352786074014132853, 11.82975374775973850453887254620, 13.41615861849473507220117410601
