| L(s) = 1 | + 6.12e4·2-s − 4.30e7·3-s − 4.83e9·4-s + 2.12e11·5-s − 2.63e12·6-s + 1.58e14·7-s − 8.22e14·8-s + 1.85e15·9-s + 1.30e16·10-s − 2.62e17·11-s + 2.08e17·12-s − 3.56e18·13-s + 9.69e18·14-s − 9.14e18·15-s − 8.85e18·16-s + 1.73e19·17-s + 1.13e20·18-s − 5.11e20·19-s − 1.02e21·20-s − 6.81e21·21-s − 1.61e22·22-s − 3.17e22·23-s + 3.54e22·24-s − 7.13e22·25-s − 2.18e23·26-s − 7.97e22·27-s − 7.65e23·28-s + ⋯ |
| L(s) = 1 | + 0.661·2-s − 0.577·3-s − 0.562·4-s + 0.622·5-s − 0.381·6-s + 1.79·7-s − 1.03·8-s + 0.333·9-s + 0.411·10-s − 1.72·11-s + 0.325·12-s − 1.48·13-s + 1.18·14-s − 0.359·15-s − 0.120·16-s + 0.0862·17-s + 0.220·18-s − 0.406·19-s − 0.350·20-s − 1.03·21-s − 1.14·22-s − 1.07·23-s + 0.596·24-s − 0.612·25-s − 0.982·26-s − 0.192·27-s − 1.01·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 4.30e7T \) |
| good | 2 | \( 1 - 6.12e4T + 8.58e9T^{2} \) |
| 5 | \( 1 - 2.12e11T + 1.16e23T^{2} \) |
| 7 | \( 1 - 1.58e14T + 7.73e27T^{2} \) |
| 11 | \( 1 + 2.62e17T + 2.32e34T^{2} \) |
| 13 | \( 1 + 3.56e18T + 5.75e36T^{2} \) |
| 17 | \( 1 - 1.73e19T + 4.02e40T^{2} \) |
| 19 | \( 1 + 5.11e20T + 1.58e42T^{2} \) |
| 23 | \( 1 + 3.17e22T + 8.65e44T^{2} \) |
| 29 | \( 1 - 1.64e23T + 1.81e48T^{2} \) |
| 31 | \( 1 - 1.52e24T + 1.64e49T^{2} \) |
| 37 | \( 1 + 4.62e25T + 5.63e51T^{2} \) |
| 41 | \( 1 + 3.97e26T + 1.66e53T^{2} \) |
| 43 | \( 1 + 3.64e26T + 8.02e53T^{2} \) |
| 47 | \( 1 + 2.74e27T + 1.51e55T^{2} \) |
| 53 | \( 1 - 3.12e28T + 7.96e56T^{2} \) |
| 59 | \( 1 + 9.28e28T + 2.74e58T^{2} \) |
| 61 | \( 1 + 8.67e28T + 8.23e58T^{2} \) |
| 67 | \( 1 - 4.18e29T + 1.82e60T^{2} \) |
| 71 | \( 1 + 4.44e30T + 1.23e61T^{2} \) |
| 73 | \( 1 - 1.71e30T + 3.08e61T^{2} \) |
| 79 | \( 1 - 2.50e31T + 4.18e62T^{2} \) |
| 83 | \( 1 - 7.18e31T + 2.13e63T^{2} \) |
| 89 | \( 1 + 1.21e32T + 2.13e64T^{2} \) |
| 97 | \( 1 - 5.67e32T + 3.65e65T^{2} \) |
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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
−17.43837582324422302370154741095, −15.00328542007060913660329522520, −13.66674508064452252117834308574, −12.06394838807788824159713110598, −10.21922508713911823311475238587, −7.988815170797356720565846935624, −5.43460478094096448163200830958, −4.72396043192429427116059972957, −2.14786426208390986970260053324, 0,
2.14786426208390986970260053324, 4.72396043192429427116059972957, 5.43460478094096448163200830958, 7.988815170797356720565846935624, 10.21922508713911823311475238587, 12.06394838807788824159713110598, 13.66674508064452252117834308574, 15.00328542007060913660329522520, 17.43837582324422302370154741095
