| L(s) = 1 | − 7.04·2-s + 86.5·3-s − 78.3·4-s − 609.·6-s + 609.·7-s + 1.45e3·8-s + 5.29e3·9-s − 454.·11-s − 6.77e3·12-s − 1.19e4·13-s − 4.29e3·14-s − 223.·16-s − 1.97e4·17-s − 3.73e4·18-s + 2.44e4·19-s + 5.27e4·21-s + 3.20e3·22-s − 4.83e3·23-s + 1.25e5·24-s + 8.41e4·26-s + 2.69e5·27-s − 4.77e4·28-s − 2.18e5·29-s + 1.27e5·31-s − 1.84e5·32-s − 3.93e4·33-s + 1.39e5·34-s + ⋯ |
| L(s) = 1 | − 0.622·2-s + 1.85·3-s − 0.611·4-s − 1.15·6-s + 0.671·7-s + 1.00·8-s + 2.42·9-s − 0.102·11-s − 1.13·12-s − 1.50·13-s − 0.418·14-s − 0.0136·16-s − 0.975·17-s − 1.50·18-s + 0.817·19-s + 1.24·21-s + 0.0641·22-s − 0.0828·23-s + 1.85·24-s + 0.938·26-s + 2.63·27-s − 0.410·28-s − 1.66·29-s + 0.771·31-s − 0.995·32-s − 0.190·33-s + 0.607·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 7.04T + 128T^{2} \) |
| 3 | \( 1 - 86.5T + 2.18e3T^{2} \) |
| 7 | \( 1 - 609.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 454.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.19e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.97e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.44e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.83e3T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.18e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.27e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.05e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.20e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.30e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.60e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.90e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.39e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.70e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.71e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.96e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.17e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.88e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.46e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.05e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 9.27e6T + 8.07e13T^{2} \) |
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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
−9.200031880577446370419408797557, −8.030138208133910709126738474531, −7.890165474005591036695124538752, −6.93784769853318043688154082301, −4.98610515305647065711029682099, −4.38475072410793859379693233871, −3.26114269237894626583238889885, −2.20738502119819760104402344220, −1.44430433625763010397455615213, 0,
1.44430433625763010397455615213, 2.20738502119819760104402344220, 3.26114269237894626583238889885, 4.38475072410793859379693233871, 4.98610515305647065711029682099, 6.93784769853318043688154082301, 7.890165474005591036695124538752, 8.030138208133910709126738474531, 9.200031880577446370419408797557
