| L(s) = 1 | − 2-s + 4-s − 1.23·5-s − 7-s − 8-s + 1.23·10-s + 5.23·11-s − 3.23·13-s + 14-s + 16-s + 5.70·17-s + 19-s − 1.23·20-s − 5.23·22-s + 0.763·23-s − 3.47·25-s + 3.23·26-s − 28-s − 0.472·29-s − 2·31-s − 32-s − 5.70·34-s + 1.23·35-s − 1.52·37-s − 38-s + 1.23·40-s + 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.552·5-s − 0.377·7-s − 0.353·8-s + 0.390·10-s + 1.57·11-s − 0.897·13-s + 0.267·14-s + 0.250·16-s + 1.38·17-s + 0.229·19-s − 0.276·20-s − 1.11·22-s + 0.159·23-s − 0.694·25-s + 0.634·26-s − 0.188·28-s − 0.0876·29-s − 0.359·31-s − 0.176·32-s − 0.978·34-s + 0.208·35-s − 0.251·37-s − 0.162·38-s + 0.195·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.102915928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.102915928\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 1.23T + 5T^{2} \) |
| 11 | \( 1 - 5.23T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 - 5.70T + 17T^{2} \) |
| 23 | \( 1 - 0.763T + 23T^{2} \) |
| 29 | \( 1 + 0.472T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 1.52T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8.94T + 43T^{2} \) |
| 47 | \( 1 + 8.94T + 47T^{2} \) |
| 53 | \( 1 - 3.52T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 8.47T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 - 1.52T + 71T^{2} \) |
| 73 | \( 1 + 3.52T + 73T^{2} \) |
| 79 | \( 1 - 5.23T + 79T^{2} \) |
| 83 | \( 1 + 10T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 9.23T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.991295892472330142328999481590, −8.225012384596411764012013771353, −7.44464356226274058360973970324, −6.87484897037954637882134003639, −6.03441320911308192521077366486, −5.07228105561494212680214631170, −3.87763298788434150674352282358, −3.27963107636739416584393614623, −1.94437144593608726573366630229, −0.75755249820635417138551374004,
0.75755249820635417138551374004, 1.94437144593608726573366630229, 3.27963107636739416584393614623, 3.87763298788434150674352282358, 5.07228105561494212680214631170, 6.03441320911308192521077366486, 6.87484897037954637882134003639, 7.44464356226274058360973970324, 8.225012384596411764012013771353, 8.991295892472330142328999481590
