| L(s) = 1 | + 0.422·2-s + 3-s − 1.82·4-s − 0.274·5-s + 0.422·6-s − 5.26·7-s − 1.61·8-s + 9-s − 0.116·10-s + 2.24·11-s − 1.82·12-s + 2.71·13-s − 2.22·14-s − 0.274·15-s + 2.95·16-s − 17-s + 0.422·18-s + 3.58·19-s + 0.500·20-s − 5.26·21-s + 0.950·22-s − 4.99·23-s − 1.61·24-s − 4.92·25-s + 1.14·26-s + 27-s + 9.59·28-s + ⋯ |
| L(s) = 1 | + 0.298·2-s + 0.577·3-s − 0.910·4-s − 0.122·5-s + 0.172·6-s − 1.99·7-s − 0.571·8-s + 0.333·9-s − 0.0367·10-s + 0.677·11-s − 0.525·12-s + 0.753·13-s − 0.595·14-s − 0.0709·15-s + 0.739·16-s − 0.242·17-s + 0.0996·18-s + 0.822·19-s + 0.111·20-s − 1.14·21-s + 0.202·22-s − 1.04·23-s − 0.329·24-s − 0.984·25-s + 0.225·26-s + 0.192·27-s + 1.81·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 - 0.422T + 2T^{2} \) |
| 5 | \( 1 + 0.274T + 5T^{2} \) |
| 7 | \( 1 + 5.26T + 7T^{2} \) |
| 11 | \( 1 - 2.24T + 11T^{2} \) |
| 13 | \( 1 - 2.71T + 13T^{2} \) |
| 19 | \( 1 - 3.58T + 19T^{2} \) |
| 23 | \( 1 + 4.99T + 23T^{2} \) |
| 29 | \( 1 + 3.94T + 29T^{2} \) |
| 31 | \( 1 - 7.05T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 7.21T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 + 4.05T + 47T^{2} \) |
| 53 | \( 1 + 1.06T + 53T^{2} \) |
| 59 | \( 1 + 7.42T + 59T^{2} \) |
| 61 | \( 1 - 1.49T + 61T^{2} \) |
| 67 | \( 1 - 0.290T + 67T^{2} \) |
| 71 | \( 1 - 7.85T + 71T^{2} \) |
| 73 | \( 1 + 9.79T + 73T^{2} \) |
| 79 | \( 1 - 2.79T + 79T^{2} \) |
| 83 | \( 1 + 2.71T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
| 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
−7.61150525661135012210966047923, −6.56354653174500197150006507448, −6.17178968561465587924284996268, −5.53566186313727965884325391939, −4.27567635777198368565624244389, −3.89345771803350739315918942347, −3.29856831527203427261859591190, −2.56643076579240850155231938475, −1.11116432799895147395346002334, 0,
1.11116432799895147395346002334, 2.56643076579240850155231938475, 3.29856831527203427261859591190, 3.89345771803350739315918942347, 4.27567635777198368565624244389, 5.53566186313727965884325391939, 6.17178968561465587924284996268, 6.56354653174500197150006507448, 7.61150525661135012210966047923
