| L(s) = 1 | + 3.14e5·2-s + 6.47e10·4-s − 2.58e12·5-s + 8.89e14·7-s + 9.58e15·8-s − 8.14e17·10-s − 2.05e18·11-s − 1.82e19·13-s + 2.79e20·14-s + 7.91e20·16-s − 3.53e21·17-s + 2.01e22·19-s − 1.67e23·20-s − 6.46e23·22-s − 2.51e23·23-s + 3.77e24·25-s − 5.73e24·26-s + 5.76e25·28-s + 8.93e24·29-s − 1.94e26·31-s − 8.00e25·32-s − 1.11e27·34-s − 2.29e27·35-s − 2.17e27·37-s + 6.34e27·38-s − 2.47e28·40-s + 7.70e27·41-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 1.88·4-s − 1.51·5-s + 1.44·7-s + 1.50·8-s − 2.57·10-s − 1.22·11-s − 0.583·13-s + 2.45·14-s + 0.670·16-s − 1.03·17-s + 0.844·19-s − 2.85·20-s − 2.07·22-s − 0.371·23-s + 1.29·25-s − 0.991·26-s + 2.72·28-s + 0.228·29-s − 1.55·31-s − 0.365·32-s − 1.76·34-s − 2.18·35-s − 0.783·37-s + 1.43·38-s − 2.28·40-s + 0.460·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(36-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+35/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 2 | \( 1 - 3.14e5T + 3.43e10T^{2} \) |
| 5 | \( 1 + 2.58e12T + 2.91e24T^{2} \) |
| 7 | \( 1 - 8.89e14T + 3.78e29T^{2} \) |
| 11 | \( 1 + 2.05e18T + 2.81e36T^{2} \) |
| 13 | \( 1 + 1.82e19T + 9.72e38T^{2} \) |
| 17 | \( 1 + 3.53e21T + 1.16e43T^{2} \) |
| 19 | \( 1 - 2.01e22T + 5.70e44T^{2} \) |
| 23 | \( 1 + 2.51e23T + 4.57e47T^{2} \) |
| 29 | \( 1 - 8.93e24T + 1.52e51T^{2} \) |
| 31 | \( 1 + 1.94e26T + 1.57e52T^{2} \) |
| 37 | \( 1 + 2.17e27T + 7.71e54T^{2} \) |
| 41 | \( 1 - 7.70e27T + 2.80e56T^{2} \) |
| 43 | \( 1 + 2.60e28T + 1.48e57T^{2} \) |
| 47 | \( 1 + 3.33e29T + 3.33e58T^{2} \) |
| 53 | \( 1 + 1.01e30T + 2.23e60T^{2} \) |
| 59 | \( 1 + 1.91e30T + 9.54e61T^{2} \) |
| 61 | \( 1 - 7.04e30T + 3.06e62T^{2} \) |
| 67 | \( 1 - 1.18e32T + 8.17e63T^{2} \) |
| 71 | \( 1 - 2.78e32T + 6.22e64T^{2} \) |
| 73 | \( 1 + 3.82e32T + 1.64e65T^{2} \) |
| 79 | \( 1 + 1.32e33T + 2.61e66T^{2} \) |
| 83 | \( 1 - 5.69e33T + 1.47e67T^{2} \) |
| 89 | \( 1 + 1.72e34T + 1.69e68T^{2} \) |
| 97 | \( 1 - 4.83e34T + 3.44e69T^{2} \) |
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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
−12.83433756118411703679080166816, −11.67336126899890400812698192704, −11.01419019109018840326510526825, −8.121427916179669358803937093381, −7.19382318349901416581444860258, −5.20915954088085506961702630813, −4.56768567219859159441332862477, −3.38243961231478284911283211613, −2.03556701936118123398412747345, 0,
2.03556701936118123398412747345, 3.38243961231478284911283211613, 4.56768567219859159441332862477, 5.20915954088085506961702630813, 7.19382318349901416581444860258, 8.121427916179669358803937093381, 11.01419019109018840326510526825, 11.67336126899890400812698192704, 12.83433756118411703679080166816
