| L(s) = 1 | − 0.265·2-s − 2.74·3-s − 1.92·4-s − 5-s + 0.728·6-s − 7-s + 1.04·8-s + 4.53·9-s + 0.265·10-s + 1.08·11-s + 5.29·12-s + 4.78·13-s + 0.265·14-s + 2.74·15-s + 3.58·16-s − 4.67·17-s − 1.20·18-s − 5.44·19-s + 1.92·20-s + 2.74·21-s − 0.287·22-s + 0.541·23-s − 2.86·24-s + 25-s − 1.26·26-s − 4.22·27-s + 1.92·28-s + ⋯ |
| L(s) = 1 | − 0.187·2-s − 1.58·3-s − 0.964·4-s − 0.447·5-s + 0.297·6-s − 0.377·7-s + 0.368·8-s + 1.51·9-s + 0.0839·10-s + 0.326·11-s + 1.52·12-s + 1.32·13-s + 0.0709·14-s + 0.708·15-s + 0.895·16-s − 1.13·17-s − 0.284·18-s − 1.24·19-s + 0.431·20-s + 0.599·21-s − 0.0613·22-s + 0.113·23-s − 0.584·24-s + 0.200·25-s − 0.249·26-s − 0.812·27-s + 0.364·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 + 0.265T + 2T^{2} \) |
| 3 | \( 1 + 2.74T + 3T^{2} \) |
| 11 | \( 1 - 1.08T + 11T^{2} \) |
| 13 | \( 1 - 4.78T + 13T^{2} \) |
| 17 | \( 1 + 4.67T + 17T^{2} \) |
| 19 | \( 1 + 5.44T + 19T^{2} \) |
| 23 | \( 1 - 0.541T + 23T^{2} \) |
| 29 | \( 1 + 5.09T + 29T^{2} \) |
| 31 | \( 1 + 0.376T + 31T^{2} \) |
| 37 | \( 1 + 8.09T + 37T^{2} \) |
| 41 | \( 1 - 4.76T + 41T^{2} \) |
| 43 | \( 1 + 0.312T + 43T^{2} \) |
| 47 | \( 1 + 4.13T + 47T^{2} \) |
| 53 | \( 1 - 4.89T + 53T^{2} \) |
| 59 | \( 1 + 0.913T + 59T^{2} \) |
| 61 | \( 1 - 2.68T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 6.31T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 2.84T + 79T^{2} \) |
| 83 | \( 1 - 1.24T + 83T^{2} \) |
| 89 | \( 1 - 1.44T + 89T^{2} \) |
| 97 | \( 1 - 5.71T + 97T^{2} \) |
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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
−7.30670130514439003326338755735, −6.63240837994096993414383996207, −6.09979934813007680490211099954, −5.44585424728220969526391290345, −4.66293418222289994132963679409, −4.08802693168085551239228415351, −3.50623740340251173387676131021, −1.87763561471692965846272471241, −0.793621857931501889627027264736, 0,
0.793621857931501889627027264736, 1.87763561471692965846272471241, 3.50623740340251173387676131021, 4.08802693168085551239228415351, 4.66293418222289994132963679409, 5.44585424728220969526391290345, 6.09979934813007680490211099954, 6.63240837994096993414383996207, 7.30670130514439003326338755735
