| L(s) = 1 | + 2.19·2-s + 2.81·4-s + 5-s + 4.33·7-s + 1.78·8-s + 2.19·10-s − 11-s − 2.66·13-s + 9.51·14-s − 1.70·16-s + 4.96·17-s + 19-s + 2.81·20-s − 2.19·22-s + 1.79·23-s + 25-s − 5.84·26-s + 12.2·28-s + 4.47·29-s − 2.66·31-s − 7.32·32-s + 10.8·34-s + 4.33·35-s + 8.67·37-s + 2.19·38-s + 1.78·40-s − 6.52·41-s + ⋯ |
| L(s) = 1 | + 1.55·2-s + 1.40·4-s + 0.447·5-s + 1.63·7-s + 0.631·8-s + 0.693·10-s − 0.301·11-s − 0.739·13-s + 2.54·14-s − 0.427·16-s + 1.20·17-s + 0.229·19-s + 0.629·20-s − 0.467·22-s + 0.375·23-s + 0.200·25-s − 1.14·26-s + 2.30·28-s + 0.831·29-s − 0.478·31-s − 1.29·32-s + 1.86·34-s + 0.733·35-s + 1.42·37-s + 0.355·38-s + 0.282·40-s − 1.01·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.188328310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.188328310\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.19T + 2T^{2} \) |
| 7 | \( 1 - 4.33T + 7T^{2} \) |
| 13 | \( 1 + 2.66T + 13T^{2} \) |
| 17 | \( 1 - 4.96T + 17T^{2} \) |
| 23 | \( 1 - 1.79T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 2.66T + 31T^{2} \) |
| 37 | \( 1 - 8.67T + 37T^{2} \) |
| 41 | \( 1 + 6.52T + 41T^{2} \) |
| 43 | \( 1 - 9.86T + 43T^{2} \) |
| 47 | \( 1 - 4.40T + 47T^{2} \) |
| 53 | \( 1 + 7.26T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 6.44T + 61T^{2} \) |
| 67 | \( 1 + 4.25T + 67T^{2} \) |
| 71 | \( 1 + 0.847T + 71T^{2} \) |
| 73 | \( 1 - 4.86T + 73T^{2} \) |
| 79 | \( 1 - 6.17T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
| 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
−7.68470762520093874151831755406, −6.87597697656735611196250336762, −5.98573071637592166676138275547, −5.47288971346105076246814887470, −4.84810587448918389174739945458, −4.53569099806195177476319464654, −3.54075768482808264133664270199, −2.71714078089946740602265495141, −2.06170561310085388279213505239, −1.07760493642242190139532488866,
1.07760493642242190139532488866, 2.06170561310085388279213505239, 2.71714078089946740602265495141, 3.54075768482808264133664270199, 4.53569099806195177476319464654, 4.84810587448918389174739945458, 5.47288971346105076246814887470, 5.98573071637592166676138275547, 6.87597697656735611196250336762, 7.68470762520093874151831755406
