| L(s) = 1 | + 2.66·2-s − 3-s + 5.08·4-s − 2.66·6-s + 0.272·7-s + 8.22·8-s + 9-s − 2.08·11-s − 5.08·12-s + 2.57·13-s + 0.725·14-s + 11.7·16-s − 2.57·17-s + 2.66·18-s + 7.07·19-s − 0.272·21-s − 5.56·22-s − 0.272·23-s − 8.22·24-s + 6.86·26-s − 27-s + 1.38·28-s + 5.74·29-s + 2.73·31-s + 14.7·32-s + 2.08·33-s − 6.86·34-s + ⋯ |
| L(s) = 1 | + 1.88·2-s − 0.577·3-s + 2.54·4-s − 1.08·6-s + 0.103·7-s + 2.90·8-s + 0.333·9-s − 0.630·11-s − 1.46·12-s + 0.714·13-s + 0.193·14-s + 2.93·16-s − 0.624·17-s + 0.627·18-s + 1.62·19-s − 0.0594·21-s − 1.18·22-s − 0.0568·23-s − 1.67·24-s + 1.34·26-s − 0.192·27-s + 0.262·28-s + 1.06·29-s + 0.491·31-s + 2.61·32-s + 0.363·33-s − 1.17·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.792642133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.792642133\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 2.66T + 2T^{2} \) |
| 7 | \( 1 - 0.272T + 7T^{2} \) |
| 11 | \( 1 + 2.08T + 11T^{2} \) |
| 13 | \( 1 - 2.57T + 13T^{2} \) |
| 17 | \( 1 + 2.57T + 17T^{2} \) |
| 19 | \( 1 - 7.07T + 19T^{2} \) |
| 23 | \( 1 + 0.272T + 23T^{2} \) |
| 29 | \( 1 - 5.74T + 29T^{2} \) |
| 31 | \( 1 - 2.73T + 31T^{2} \) |
| 37 | \( 1 - 0.0895T + 37T^{2} \) |
| 41 | \( 1 - 2.81T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 53 | \( 1 + 8.14T + 53T^{2} \) |
| 59 | \( 1 + 8.05T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 - 4.04T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 + 0.736T + 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 6.13T + 89T^{2} \) |
| 97 | \( 1 - 3.51T + 97T^{2} \) |
| 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
−8.205513616408117966416109695557, −7.53818116522159025352504127828, −6.63639957140309499241145695037, −6.22762986963267373261690914554, −5.27068371050072260802313303764, −4.96395891796512792192591197991, −4.04251661731410521356745563347, −3.24437974084311022002711105974, −2.40997529185232950368929541468, −1.19997301710274277429738812556,
1.19997301710274277429738812556, 2.40997529185232950368929541468, 3.24437974084311022002711105974, 4.04251661731410521356745563347, 4.96395891796512792192591197991, 5.27068371050072260802313303764, 6.22762986963267373261690914554, 6.63639957140309499241145695037, 7.53818116522159025352504127828, 8.205513616408117966416109695557
