| L(s) = 1 | − 505.·2-s + 1.24e5·4-s − 3.90e5·5-s − 2.08e7·7-s + 3.25e6·8-s + 1.97e8·10-s − 4.88e8·11-s + 2.14e9·13-s + 1.05e10·14-s − 1.79e10·16-s + 4.75e9·17-s + 5.53e10·19-s − 4.86e10·20-s + 2.47e11·22-s − 4.55e11·23-s + 1.52e11·25-s − 1.08e12·26-s − 2.59e12·28-s − 4.05e12·29-s + 7.66e12·31-s + 8.66e12·32-s − 2.40e12·34-s + 8.14e12·35-s + 6.26e12·37-s − 2.80e13·38-s − 1.27e12·40-s + 4.51e13·41-s + ⋯ |
| L(s) = 1 | − 1.39·2-s + 0.950·4-s − 0.447·5-s − 1.36·7-s + 0.0685·8-s + 0.624·10-s − 0.687·11-s + 0.728·13-s + 1.90·14-s − 1.04·16-s + 0.165·17-s + 0.748·19-s − 0.425·20-s + 0.959·22-s − 1.21·23-s + 0.200·25-s − 1.01·26-s − 1.29·28-s − 1.50·29-s + 1.61·31-s + 1.39·32-s − 0.231·34-s + 0.611·35-s + 0.293·37-s − 1.04·38-s − 0.0306·40-s + 0.882·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 3.90e5T \) |
| good | 2 | \( 1 + 505.T + 1.31e5T^{2} \) |
| 7 | \( 1 + 2.08e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 4.88e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 2.14e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 4.75e9T + 8.27e20T^{2} \) |
| 19 | \( 1 - 5.53e10T + 5.48e21T^{2} \) |
| 23 | \( 1 + 4.55e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 4.05e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 7.66e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 6.26e12T + 4.56e26T^{2} \) |
| 41 | \( 1 - 4.51e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 9.60e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 2.24e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 2.28e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 6.92e14T + 1.27e30T^{2} \) |
| 61 | \( 1 - 5.53e14T + 2.24e30T^{2} \) |
| 67 | \( 1 - 5.24e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 1.98e14T + 2.96e31T^{2} \) |
| 73 | \( 1 - 2.70e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 2.50e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 9.33e15T + 4.21e32T^{2} \) |
| 89 | \( 1 + 6.43e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 4.42e16T + 5.95e33T^{2} \) |
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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
−11.30263556540756635305583571406, −10.16436494789826995867289726714, −9.395508532768318112739648425510, −8.199222682937474220462608409216, −7.23761692796960307308902252772, −5.92305619378658191544209613295, −3.94972567101387432869832594964, −2.56728737888046286158787052334, −0.934467789083203525937592256854, 0,
0.934467789083203525937592256854, 2.56728737888046286158787052334, 3.94972567101387432869832594964, 5.92305619378658191544209613295, 7.23761692796960307308902252772, 8.199222682937474220462608409216, 9.395508532768318112739648425510, 10.16436494789826995867289726714, 11.30263556540756635305583571406
