| L(s) = 1 | − 7.19e4·2-s − 1.37e8·3-s − 3.41e9·4-s + 9.89e12·6-s + 1.07e14·7-s + 8.63e14·8-s + 1.33e16·9-s − 2.33e17·11-s + 4.69e17·12-s + 2.28e18·13-s − 7.74e18·14-s − 3.27e19·16-s + 2.09e20·17-s − 9.61e20·18-s + 6.01e20·19-s − 1.48e22·21-s + 1.67e22·22-s − 4.84e22·23-s − 1.18e23·24-s − 1.64e23·26-s − 1.07e24·27-s − 3.67e23·28-s − 8.15e22·29-s − 1.49e24·31-s − 5.05e24·32-s + 3.21e25·33-s − 1.50e25·34-s + ⋯ |
| L(s) = 1 | − 0.776·2-s − 1.84·3-s − 0.397·4-s + 1.43·6-s + 1.22·7-s + 1.08·8-s + 2.40·9-s − 1.53·11-s + 0.733·12-s + 0.951·13-s − 0.950·14-s − 0.444·16-s + 1.04·17-s − 1.86·18-s + 0.478·19-s − 2.25·21-s + 1.18·22-s − 1.64·23-s − 2.00·24-s − 0.738·26-s − 2.59·27-s − 0.486·28-s − 0.0605·29-s − 0.370·31-s − 0.739·32-s + 2.82·33-s − 0.811·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(17)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{35}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 7.19e4T + 8.58e9T^{2} \) |
| 3 | \( 1 + 1.37e8T + 5.55e15T^{2} \) |
| 7 | \( 1 - 1.07e14T + 7.73e27T^{2} \) |
| 11 | \( 1 + 2.33e17T + 2.32e34T^{2} \) |
| 13 | \( 1 - 2.28e18T + 5.75e36T^{2} \) |
| 17 | \( 1 - 2.09e20T + 4.02e40T^{2} \) |
| 19 | \( 1 - 6.01e20T + 1.58e42T^{2} \) |
| 23 | \( 1 + 4.84e22T + 8.65e44T^{2} \) |
| 29 | \( 1 + 8.15e22T + 1.81e48T^{2} \) |
| 31 | \( 1 + 1.49e24T + 1.64e49T^{2} \) |
| 37 | \( 1 + 1.78e25T + 5.63e51T^{2} \) |
| 41 | \( 1 + 1.94e26T + 1.66e53T^{2} \) |
| 43 | \( 1 - 9.48e26T + 8.02e53T^{2} \) |
| 47 | \( 1 - 6.29e26T + 1.51e55T^{2} \) |
| 53 | \( 1 - 8.56e27T + 7.96e56T^{2} \) |
| 59 | \( 1 - 5.43e27T + 2.74e58T^{2} \) |
| 61 | \( 1 - 1.79e29T + 8.23e58T^{2} \) |
| 67 | \( 1 - 3.33e29T + 1.82e60T^{2} \) |
| 71 | \( 1 + 3.44e30T + 1.23e61T^{2} \) |
| 73 | \( 1 - 7.11e30T + 3.08e61T^{2} \) |
| 79 | \( 1 + 1.57e31T + 4.18e62T^{2} \) |
| 83 | \( 1 + 3.45e31T + 2.13e63T^{2} \) |
| 89 | \( 1 + 2.35e32T + 2.13e64T^{2} \) |
| 97 | \( 1 - 3.55e32T + 3.65e65T^{2} \) |
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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
−10.65891986352172879648712163672, −9.986343695196854203295168558190, −8.207273525412332473359862842731, −7.44937724225136466503435223591, −5.73217215316518328320498515581, −5.16239706309145318280976204772, −4.11198989324844633694985721018, −1.72620562709065477085507548982, −0.895608330030174634497340690715, 0,
0.895608330030174634497340690715, 1.72620562709065477085507548982, 4.11198989324844633694985721018, 5.16239706309145318280976204772, 5.73217215316518328320498515581, 7.44937724225136466503435223591, 8.207273525412332473359862842731, 9.986343695196854203295168558190, 10.65891986352172879648712163672
