| L(s) = 1 | − 364.·2-s + 2.09e4·3-s + 1.87e3·4-s + 3.90e5·5-s − 7.64e6·6-s − 1.29e7·7-s + 4.71e7·8-s + 3.11e8·9-s − 1.42e8·10-s + 7.40e8·11-s + 3.92e7·12-s + 5.88e8·13-s + 4.72e9·14-s + 8.19e9·15-s − 1.74e10·16-s + 1.97e10·17-s − 1.13e11·18-s + 1.15e11·19-s + 7.31e8·20-s − 2.72e11·21-s − 2.69e11·22-s − 1.32e10·23-s + 9.88e11·24-s + 1.52e11·25-s − 2.14e11·26-s + 3.81e12·27-s − 2.42e10·28-s + ⋯ |
| L(s) = 1 | − 1.00·2-s + 1.84·3-s + 0.0142·4-s + 0.447·5-s − 1.85·6-s − 0.850·7-s + 0.992·8-s + 2.40·9-s − 0.450·10-s + 1.04·11-s + 0.0263·12-s + 0.200·13-s + 0.856·14-s + 0.825·15-s − 1.01·16-s + 0.685·17-s − 2.42·18-s + 1.56·19-s + 0.00639·20-s − 1.56·21-s − 1.04·22-s − 0.0353·23-s + 1.83·24-s + 0.200·25-s − 0.201·26-s + 2.59·27-s − 0.0121·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(1.877124152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.877124152\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 3.90e5T \) |
| good | 2 | \( 1 + 364.T + 1.31e5T^{2} \) |
| 3 | \( 1 - 2.09e4T + 1.29e8T^{2} \) |
| 7 | \( 1 + 1.29e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 7.40e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 5.88e8T + 8.65e18T^{2} \) |
| 17 | \( 1 - 1.97e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 1.15e11T + 5.48e21T^{2} \) |
| 23 | \( 1 + 1.32e10T + 1.41e23T^{2} \) |
| 29 | \( 1 + 3.75e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 5.77e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 4.16e12T + 4.56e26T^{2} \) |
| 41 | \( 1 + 3.86e12T + 2.61e27T^{2} \) |
| 43 | \( 1 - 2.01e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 1.74e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 2.15e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 1.60e15T + 1.27e30T^{2} \) |
| 61 | \( 1 - 1.25e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 2.01e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 3.92e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 5.25e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 8.37e15T + 1.81e32T^{2} \) |
| 83 | \( 1 - 3.88e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 2.17e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 3.76e16T + 5.95e33T^{2} \) |
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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
−19.43793374489451203535034505589, −18.37986748330539665914971327298, −16.36333002850223249810572669030, −14.42265921724215901422088248354, −13.25419151005070291404008829589, −9.744181791137771270821176932431, −9.096276102271981844063281453576, −7.45826875672673278899382571089, −3.49955013173593770312773893330, −1.48746059537090573453486945999,
1.48746059537090573453486945999, 3.49955013173593770312773893330, 7.45826875672673278899382571089, 9.096276102271981844063281453576, 9.744181791137771270821176932431, 13.25419151005070291404008829589, 14.42265921724215901422088248354, 16.36333002850223249810572669030, 18.37986748330539665914971327298, 19.43793374489451203535034505589
