| L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s + 0.381·11-s + 4.61·13-s + 2·14-s + 16-s − 2.61·17-s + 3.23·19-s + 0.381·22-s − 3.61·23-s + 4.61·26-s + 2·28-s + 9.09·29-s − 4.85·31-s + 32-s − 2.61·34-s − 0.0901·37-s + 3.23·38-s − 8.94·41-s + 8.85·43-s + 0.381·44-s − 3.61·46-s + 3.09·47-s − 3·49-s + 4.61·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.755·7-s + 0.353·8-s + 0.115·11-s + 1.28·13-s + 0.534·14-s + 0.250·16-s − 0.634·17-s + 0.742·19-s + 0.0814·22-s − 0.754·23-s + 0.905·26-s + 0.377·28-s + 1.68·29-s − 0.871·31-s + 0.176·32-s − 0.448·34-s − 0.0148·37-s + 0.524·38-s − 1.39·41-s + 1.35·43-s + 0.0575·44-s − 0.533·46-s + 0.450·47-s − 0.428·49-s + 0.640·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.334328393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.334328393\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 0.381T + 11T^{2} \) |
| 13 | \( 1 - 4.61T + 13T^{2} \) |
| 17 | \( 1 + 2.61T + 17T^{2} \) |
| 19 | \( 1 - 3.23T + 19T^{2} \) |
| 23 | \( 1 + 3.61T + 23T^{2} \) |
| 29 | \( 1 - 9.09T + 29T^{2} \) |
| 31 | \( 1 + 4.85T + 31T^{2} \) |
| 37 | \( 1 + 0.0901T + 37T^{2} \) |
| 41 | \( 1 + 8.94T + 41T^{2} \) |
| 43 | \( 1 - 8.85T + 43T^{2} \) |
| 47 | \( 1 - 3.09T + 47T^{2} \) |
| 53 | \( 1 - 3.23T + 53T^{2} \) |
| 59 | \( 1 - 4.32T + 59T^{2} \) |
| 61 | \( 1 - 3.70T + 61T^{2} \) |
| 67 | \( 1 - 2.90T + 67T^{2} \) |
| 71 | \( 1 + 5.23T + 71T^{2} \) |
| 73 | \( 1 - 16.1T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + 8.18T + 83T^{2} \) |
| 89 | \( 1 - 2.47T + 89T^{2} \) |
| 97 | \( 1 + 6.94T + 97T^{2} \) |
| 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
−8.804181025494535824309257002516, −8.301684841039559083689379837758, −7.40022662110918957063462798508, −6.56531911520402224185498452132, −5.82884046545610601773012817457, −5.01134447396947593059756516889, −4.18757048351643343488194050060, −3.40607461160931023396936957202, −2.24279137063684823915149813983, −1.17826919220243548563301951406,
1.17826919220243548563301951406, 2.24279137063684823915149813983, 3.40607461160931023396936957202, 4.18757048351643343488194050060, 5.01134447396947593059756516889, 5.82884046545610601773012817457, 6.56531911520402224185498452132, 7.40022662110918957063462798508, 8.301684841039559083689379837758, 8.804181025494535824309257002516
