| L(s) = 1 | + 36.7·2-s − 193.·3-s + 839.·4-s − 7.11e3·6-s − 7.64e3·7-s + 1.20e4·8-s + 1.77e4·9-s − 4.83e4·11-s − 1.62e5·12-s − 1.00e5·13-s − 2.81e5·14-s + 1.29e4·16-s + 2.01e5·17-s + 6.53e5·18-s − 5.80e4·19-s + 1.48e6·21-s − 1.77e6·22-s + 1.14e6·23-s − 2.33e6·24-s − 3.69e6·26-s + 3.70e5·27-s − 6.42e6·28-s + 1.56e6·29-s − 4.10e6·31-s − 5.69e6·32-s + 9.35e6·33-s + 7.42e6·34-s + ⋯ |
| L(s) = 1 | + 1.62·2-s − 1.37·3-s + 1.63·4-s − 2.24·6-s − 1.20·7-s + 1.03·8-s + 0.902·9-s − 0.995·11-s − 2.26·12-s − 0.975·13-s − 1.95·14-s + 0.0494·16-s + 0.586·17-s + 1.46·18-s − 0.102·19-s + 1.66·21-s − 1.61·22-s + 0.850·23-s − 1.43·24-s − 1.58·26-s + 0.134·27-s − 1.97·28-s + 0.410·29-s − 0.798·31-s − 0.959·32-s + 1.37·33-s + 0.952·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 36.7T + 512T^{2} \) |
| 3 | \( 1 + 193.T + 1.96e4T^{2} \) |
| 7 | \( 1 + 7.64e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.83e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.00e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.01e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 5.80e4T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.14e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.56e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.10e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.70e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 8.52e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.56e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.58e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.56e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 2.15e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.15e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 7.69e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.95e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.40e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.92e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 7.88e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 8.40e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.35e8T + 7.60e17T^{2} \) |
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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
−14.82496506402777804016274950053, −13.16067745151898619909940083016, −12.52891015322837974386014630597, −11.43032420963384393022886203641, −10.07420247657848529942379907547, −6.97673585493175407428223721694, −5.82857712486537173133318472291, −4.84402438584490053687838584315, −2.98328825830355315463771742381, 0,
2.98328825830355315463771742381, 4.84402438584490053687838584315, 5.82857712486537173133318472291, 6.97673585493175407428223721694, 10.07420247657848529942379907547, 11.43032420963384393022886203641, 12.52891015322837974386014630597, 13.16067745151898619909940083016, 14.82496506402777804016274950053
