| L(s) = 1 | − 4.15·2-s − 14.7·4-s + 37.5·5-s − 76.4·7-s + 194.·8-s − 155.·10-s + 121·11-s + 169.·13-s + 317.·14-s − 333.·16-s + 0.875·17-s − 817.·19-s − 553.·20-s − 502.·22-s − 749.·23-s − 1.71e3·25-s − 704.·26-s + 1.12e3·28-s − 6.04e3·29-s − 1.47e3·31-s − 4.82e3·32-s − 3.63·34-s − 2.86e3·35-s − 1.58e4·37-s + 3.39e3·38-s + 7.28e3·40-s + 7.62e3·41-s + ⋯ |
| L(s) = 1 | − 0.733·2-s − 0.461·4-s + 0.671·5-s − 0.589·7-s + 1.07·8-s − 0.492·10-s + 0.301·11-s + 0.278·13-s + 0.433·14-s − 0.325·16-s + 0.000735·17-s − 0.519·19-s − 0.309·20-s − 0.221·22-s − 0.295·23-s − 0.549·25-s − 0.204·26-s + 0.272·28-s − 1.33·29-s − 0.275·31-s − 0.833·32-s − 0.000539·34-s − 0.395·35-s − 1.90·37-s + 0.381·38-s + 0.719·40-s + 0.708·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 + 4.15T + 32T^{2} \) |
| 5 | \( 1 - 37.5T + 3.12e3T^{2} \) |
| 7 | \( 1 + 76.4T + 1.68e4T^{2} \) |
| 13 | \( 1 - 169.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 0.875T + 1.41e6T^{2} \) |
| 19 | \( 1 + 817.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 749.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.04e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.47e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.58e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.62e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.82e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.28e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.17e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.16e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.40e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.69e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.75e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.04e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.21e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.19e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.46e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.38e5T + 8.58e9T^{2} \) |
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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
−12.58651233270413648155009196573, −11.03296711132702169116769874310, −9.945359298232763660747264942821, −9.272018085041959485870937312845, −8.195603649430960062568887109846, −6.78843185294373572905191810792, −5.43579015165024143836678590656, −3.81065455869255812422439185330, −1.74632912364525360646673306111, 0,
1.74632912364525360646673306111, 3.81065455869255812422439185330, 5.43579015165024143836678590656, 6.78843185294373572905191810792, 8.195603649430960062568887109846, 9.272018085041959485870937312845, 9.945359298232763660747264942821, 11.03296711132702169116769874310, 12.58651233270413648155009196573
