| L(s) = 1 | − 2.33·2-s − 3-s + 3.43·4-s + 2.33·6-s − 3.47·7-s − 3.34·8-s + 9-s − 0.434·11-s − 3.43·12-s + 4.87·13-s + 8.09·14-s + 0.926·16-s − 4.87·17-s − 2.33·18-s − 3.45·19-s + 3.47·21-s + 1.01·22-s + 3.47·23-s + 3.34·24-s − 11.3·26-s − 27-s − 11.9·28-s − 2.91·29-s + 7.23·31-s + 4.52·32-s + 0.434·33-s + 11.3·34-s + ⋯ |
| L(s) = 1 | − 1.64·2-s − 0.577·3-s + 1.71·4-s + 0.951·6-s − 1.31·7-s − 1.18·8-s + 0.333·9-s − 0.130·11-s − 0.991·12-s + 1.35·13-s + 2.16·14-s + 0.231·16-s − 1.18·17-s − 0.549·18-s − 0.792·19-s + 0.757·21-s + 0.215·22-s + 0.723·23-s + 0.682·24-s − 2.22·26-s − 0.192·27-s − 2.25·28-s − 0.541·29-s + 1.29·31-s + 0.800·32-s + 0.0756·33-s + 1.94·34-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
| \(L(1)\) |
\(\approx\) |
\(0.3482540190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3482540190\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.33T + 2T^{2} \) |
| 7 | \( 1 + 3.47T + 7T^{2} \) |
| 11 | \( 1 + 0.434T + 11T^{2} \) |
| 13 | \( 1 - 4.87T + 13T^{2} \) |
| 17 | \( 1 + 4.87T + 17T^{2} \) |
| 19 | \( 1 + 3.45T + 19T^{2} \) |
| 23 | \( 1 - 3.47T + 23T^{2} \) |
| 29 | \( 1 + 2.91T + 29T^{2} \) |
| 31 | \( 1 - 7.23T + 31T^{2} \) |
| 37 | \( 1 + 1.56T + 37T^{2} \) |
| 41 | \( 1 + 6.84T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 53 | \( 1 + 3.86T + 53T^{2} \) |
| 59 | \( 1 + 5.42T + 59T^{2} \) |
| 61 | \( 1 - 1.95T + 61T^{2} \) |
| 67 | \( 1 + 2.95T + 67T^{2} \) |
| 71 | \( 1 + 5.88T + 71T^{2} \) |
| 73 | \( 1 + 5.23T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 + 9.82T + 89T^{2} \) |
| 97 | \( 1 - 3.94T + 97T^{2} \) |
| 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
−8.859320022043438192128404424204, −7.999994498239730058312588148031, −7.07939604154490451237903821436, −6.38011034748090868863702018294, −6.21748531738690019515786213985, −4.81714856549979674932220865772, −3.74604679500074219684597002405, −2.71886234310983480412704254763, −1.60175721388978254330029561029, −0.46303540301404582632922298467,
0.46303540301404582632922298467, 1.60175721388978254330029561029, 2.71886234310983480412704254763, 3.74604679500074219684597002405, 4.81714856549979674932220865772, 6.21748531738690019515786213985, 6.38011034748090868863702018294, 7.07939604154490451237903821436, 7.999994498239730058312588148031, 8.859320022043438192128404424204
