| L(s) = 1 | − 1.48·2-s − 0.806·3-s + 0.193·4-s − 5-s + 1.19·6-s + 2.67·8-s − 2.35·9-s + 1.48·10-s + 0.962·11-s − 0.156·12-s − 6.15·13-s + 0.806·15-s − 4.35·16-s + 6.31·17-s + 3.48·18-s + 19-s − 0.193·20-s − 1.42·22-s − 4.96·23-s − 2.15·24-s + 25-s + 9.11·26-s + 4.31·27-s − 3.61·29-s − 1.19·30-s + 5.92·31-s + 1.09·32-s + ⋯ |
| L(s) = 1 | − 1.04·2-s − 0.465·3-s + 0.0969·4-s − 0.447·5-s + 0.487·6-s + 0.945·8-s − 0.783·9-s + 0.468·10-s + 0.290·11-s − 0.0451·12-s − 1.70·13-s + 0.208·15-s − 1.08·16-s + 1.53·17-s + 0.820·18-s + 0.229·19-s − 0.0433·20-s − 0.303·22-s − 1.03·23-s − 0.440·24-s + 0.200·25-s + 1.78·26-s + 0.829·27-s − 0.670·29-s − 0.217·30-s + 1.06·31-s + 0.193·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4655 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4655 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.48T + 2T^{2} \) |
| 3 | \( 1 + 0.806T + 3T^{2} \) |
| 11 | \( 1 - 0.962T + 11T^{2} \) |
| 13 | \( 1 + 6.15T + 13T^{2} \) |
| 17 | \( 1 - 6.31T + 17T^{2} \) |
| 23 | \( 1 + 4.96T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 - 5.92T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 + 6.31T + 41T^{2} \) |
| 43 | \( 1 + 4.12T + 43T^{2} \) |
| 47 | \( 1 + 3.35T + 47T^{2} \) |
| 53 | \( 1 - 1.84T + 53T^{2} \) |
| 59 | \( 1 - 6.38T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 6.73T + 67T^{2} \) |
| 71 | \( 1 + 0.775T + 71T^{2} \) |
| 73 | \( 1 + 0.387T + 73T^{2} \) |
| 79 | \( 1 + 0.836T + 79T^{2} \) |
| 83 | \( 1 - 7.03T + 83T^{2} \) |
| 89 | \( 1 + 7.08T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
| 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
−8.011915666740186733677865887489, −7.49206329860095084632674505838, −6.72121755911688056512550681256, −5.71640694741623061091823138128, −5.05225829910129643774446435658, −4.29770547917613534314364410336, −3.23835990474773098638717171396, −2.21976010588856343438611701369, −0.940908680716241389224322805766, 0,
0.940908680716241389224322805766, 2.21976010588856343438611701369, 3.23835990474773098638717171396, 4.29770547917613534314364410336, 5.05225829910129643774446435658, 5.71640694741623061091823138128, 6.72121755911688056512550681256, 7.49206329860095084632674505838, 8.011915666740186733677865887489
