| L(s) = 1 | + 2.18e5·2-s + 1.29e8·3-s + 1.34e10·4-s − 9.68e11·5-s + 2.82e13·6-s − 5.65e14·7-s − 4.57e15·8-s + 1.66e16·9-s − 2.11e17·10-s − 1.62e18·11-s + 1.73e18·12-s − 2.52e19·13-s − 1.23e20·14-s − 1.25e20·15-s − 1.46e21·16-s + 4.37e21·17-s + 3.64e21·18-s + 1.72e22·19-s − 1.29e22·20-s − 7.30e22·21-s − 3.54e23·22-s − 5.13e23·23-s − 5.91e23·24-s − 1.97e24·25-s − 5.51e24·26-s + 2.15e24·27-s − 7.57e24·28-s + ⋯ |
| L(s) = 1 | + 1.17·2-s + 0.577·3-s + 0.390·4-s − 0.567·5-s + 0.680·6-s − 0.918·7-s − 0.719·8-s + 0.333·9-s − 0.669·10-s − 0.968·11-s + 0.225·12-s − 0.809·13-s − 1.08·14-s − 0.327·15-s − 1.23·16-s + 1.28·17-s + 0.393·18-s + 0.720·19-s − 0.221·20-s − 0.530·21-s − 1.14·22-s − 0.758·23-s − 0.415·24-s − 0.677·25-s − 0.954·26-s + 0.192·27-s − 0.358·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(36-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+35/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(18)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{37}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 1.29e8T \) |
| good | 2 | \( 1 - 2.18e5T + 3.43e10T^{2} \) |
| 5 | \( 1 + 9.68e11T + 2.91e24T^{2} \) |
| 7 | \( 1 + 5.65e14T + 3.78e29T^{2} \) |
| 11 | \( 1 + 1.62e18T + 2.81e36T^{2} \) |
| 13 | \( 1 + 2.52e19T + 9.72e38T^{2} \) |
| 17 | \( 1 - 4.37e21T + 1.16e43T^{2} \) |
| 19 | \( 1 - 1.72e22T + 5.70e44T^{2} \) |
| 23 | \( 1 + 5.13e23T + 4.57e47T^{2} \) |
| 29 | \( 1 + 3.39e25T + 1.52e51T^{2} \) |
| 31 | \( 1 + 9.89e25T + 1.57e52T^{2} \) |
| 37 | \( 1 - 4.58e27T + 7.71e54T^{2} \) |
| 41 | \( 1 + 2.87e28T + 2.80e56T^{2} \) |
| 43 | \( 1 - 6.85e28T + 1.48e57T^{2} \) |
| 47 | \( 1 + 4.79e26T + 3.33e58T^{2} \) |
| 53 | \( 1 - 2.32e30T + 2.23e60T^{2} \) |
| 59 | \( 1 - 5.75e30T + 9.54e61T^{2} \) |
| 61 | \( 1 + 2.67e31T + 3.06e62T^{2} \) |
| 67 | \( 1 - 7.66e31T + 8.17e63T^{2} \) |
| 71 | \( 1 + 5.08e31T + 6.22e64T^{2} \) |
| 73 | \( 1 + 2.74e32T + 1.64e65T^{2} \) |
| 79 | \( 1 + 2.13e33T + 2.61e66T^{2} \) |
| 83 | \( 1 + 2.22e33T + 1.47e67T^{2} \) |
| 89 | \( 1 - 2.07e34T + 1.69e68T^{2} \) |
| 97 | \( 1 - 1.55e34T + 3.44e69T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.04495064671967522951730817371, −14.74964043303869700742126825607, −13.31385302566793651356660744058, −12.13806351546674382447127171779, −9.694634232283678999756323381430, −7.60188097945322164421779557895, −5.55575620762707654186801387240, −3.85875687224132159374955368759, −2.75682651962078438686755235773, 0,
2.75682651962078438686755235773, 3.85875687224132159374955368759, 5.55575620762707654186801387240, 7.60188097945322164421779557895, 9.694634232283678999756323381430, 12.13806351546674382447127171779, 13.31385302566793651356660744058, 14.74964043303869700742126825607, 16.04495064671967522951730817371
