| L(s) = 1 | − 1.74e4·2-s + 9.52e6·3-s − 2.32e8·4-s − 1.12e10·5-s − 1.66e11·6-s − 2.98e12·7-s + 1.34e13·8-s + 2.20e13·9-s + 1.96e14·10-s − 1.25e15·11-s − 2.21e15·12-s + 6.09e15·13-s + 5.20e16·14-s − 1.07e17·15-s − 1.08e17·16-s + 8.28e16·17-s − 3.84e17·18-s + 3.72e18·19-s + 2.62e18·20-s − 2.84e19·21-s + 2.19e19·22-s − 1.69e19·23-s + 1.27e20·24-s − 5.94e19·25-s − 1.06e20·26-s − 4.43e20·27-s + 6.95e20·28-s + ⋯ |
| L(s) = 1 | − 0.752·2-s + 1.14·3-s − 0.433·4-s − 0.825·5-s − 0.865·6-s − 1.66·7-s + 1.07·8-s + 0.321·9-s + 0.620·10-s − 0.998·11-s − 0.498·12-s + 0.429·13-s + 1.25·14-s − 0.948·15-s − 0.377·16-s + 0.119·17-s − 0.242·18-s + 1.06·19-s + 0.358·20-s − 1.91·21-s + 0.751·22-s − 0.304·23-s + 1.24·24-s − 0.319·25-s − 0.322·26-s − 0.779·27-s + 0.722·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(30-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+29/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(15)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 1.74e4T + 5.36e8T^{2} \) |
| 3 | \( 1 - 9.52e6T + 6.86e13T^{2} \) |
| 5 | \( 1 + 1.12e10T + 1.86e20T^{2} \) |
| 7 | \( 1 + 2.98e12T + 3.21e24T^{2} \) |
| 11 | \( 1 + 1.25e15T + 1.58e30T^{2} \) |
| 13 | \( 1 - 6.09e15T + 2.01e32T^{2} \) |
| 17 | \( 1 - 8.28e16T + 4.81e35T^{2} \) |
| 19 | \( 1 - 3.72e18T + 1.21e37T^{2} \) |
| 23 | \( 1 + 1.69e19T + 3.09e39T^{2} \) |
| 29 | \( 1 + 8.81e20T + 2.56e42T^{2} \) |
| 31 | \( 1 - 3.71e21T + 1.77e43T^{2} \) |
| 37 | \( 1 - 1.97e22T + 3.00e45T^{2} \) |
| 41 | \( 1 + 1.00e23T + 5.89e46T^{2} \) |
| 43 | \( 1 + 1.61e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + 2.61e24T + 3.09e48T^{2} \) |
| 53 | \( 1 + 1.44e25T + 1.00e50T^{2} \) |
| 59 | \( 1 - 5.87e24T + 2.26e51T^{2} \) |
| 61 | \( 1 + 6.18e25T + 5.95e51T^{2} \) |
| 67 | \( 1 - 1.29e26T + 9.04e52T^{2} \) |
| 71 | \( 1 + 3.63e26T + 4.85e53T^{2} \) |
| 73 | \( 1 - 1.45e27T + 1.08e54T^{2} \) |
| 79 | \( 1 + 3.93e27T + 1.07e55T^{2} \) |
| 83 | \( 1 - 6.61e27T + 4.50e55T^{2} \) |
| 89 | \( 1 - 6.47e27T + 3.40e56T^{2} \) |
| 97 | \( 1 - 9.52e28T + 4.13e57T^{2} \) |
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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
−25.72868657660485147030428912722, −22.91132095957644566528577857099, −19.93377438460368859215936465405, −18.82169509011267587663733755437, −15.92917042077896086436932580822, −13.38692374908591032596541219977, −9.637249782268918632286389013119, −7.949866560858724884326906739463, −3.34698244840578429721006084056, 0,
3.34698244840578429721006084056, 7.949866560858724884326906739463, 9.637249782268918632286389013119, 13.38692374908591032596541219977, 15.92917042077896086436932580822, 18.82169509011267587663733755437, 19.93377438460368859215936465405, 22.91132095957644566528577857099, 25.72868657660485147030428912722
