| L(s) = 1 | + 15.7·2-s − 69.5·3-s + 119.·4-s + 316.·5-s − 1.09e3·6-s − 1.28e3·7-s − 139.·8-s + 2.65e3·9-s + 4.97e3·10-s − 8.26e3·11-s − 8.28e3·12-s + 1.21e3·13-s − 2.02e4·14-s − 2.20e4·15-s − 1.74e4·16-s − 2.45e4·17-s + 4.16e4·18-s + 5.00e4·19-s + 3.77e4·20-s + 8.97e4·21-s − 1.29e5·22-s − 7.97e3·23-s + 9.71e3·24-s + 2.21e4·25-s + 1.90e4·26-s − 3.23e4·27-s − 1.53e5·28-s + ⋯ |
| L(s) = 1 | + 1.38·2-s − 1.48·3-s + 0.930·4-s + 1.13·5-s − 2.06·6-s − 1.42·7-s − 0.0964·8-s + 1.21·9-s + 1.57·10-s − 1.87·11-s − 1.38·12-s + 0.152·13-s − 1.97·14-s − 1.68·15-s − 1.06·16-s − 1.21·17-s + 1.68·18-s + 1.67·19-s + 1.05·20-s + 2.11·21-s − 2.60·22-s − 0.136·23-s + 0.143·24-s + 0.284·25-s + 0.212·26-s − 0.316·27-s − 1.32·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 37 | \( 1 - 5.06e4T \) |
| good | 2 | \( 1 - 15.7T + 128T^{2} \) |
| 3 | \( 1 + 69.5T + 2.18e3T^{2} \) |
| 5 | \( 1 - 316.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.28e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 8.26e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.21e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.45e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.00e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.97e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.41e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.62e5T + 2.75e10T^{2} \) |
| 41 | \( 1 + 8.78e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.66e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.95e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 5.71e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.56e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 5.75e4T + 3.14e12T^{2} \) |
| 67 | \( 1 - 8.37e4T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.07e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.36e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.41e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.49e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.29e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.05e7T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.67750986059796220763964795712, −13.13533518727867721233698454370, −12.21351882658276400929344908818, −10.78925219395778380570755801535, −9.689806212553002663919311437868, −6.68346500555374368438674420537, −5.79824857122881923920542082195, −5.00277722021699284462373057152, −2.83919697979603789766007471139, 0,
2.83919697979603789766007471139, 5.00277722021699284462373057152, 5.79824857122881923920542082195, 6.68346500555374368438674420537, 9.689806212553002663919311437868, 10.78925219395778380570755801535, 12.21351882658276400929344908818, 13.13533518727867721233698454370, 13.67750986059796220763964795712
