| L(s) = 1 | − 18.8·2-s + 75.2·3-s + 229.·4-s − 439.·5-s − 1.42e3·6-s + 380.·7-s − 1.91e3·8-s + 3.48e3·9-s + 8.30e3·10-s − 6.42e3·11-s + 1.72e4·12-s + 6.84e3·13-s − 7.18e3·14-s − 3.30e4·15-s + 6.79e3·16-s + 7.86e3·17-s − 6.58e4·18-s − 4.57e4·19-s − 1.00e5·20-s + 2.86e4·21-s + 1.21e5·22-s − 8.41e4·23-s − 1.43e5·24-s + 1.15e5·25-s − 1.29e5·26-s + 9.75e4·27-s + 8.71e4·28-s + ⋯ |
| L(s) = 1 | − 1.67·2-s + 1.61·3-s + 1.79·4-s − 1.57·5-s − 2.68·6-s + 0.419·7-s − 1.31·8-s + 1.59·9-s + 2.62·10-s − 1.45·11-s + 2.88·12-s + 0.864·13-s − 0.699·14-s − 2.53·15-s + 0.414·16-s + 0.388·17-s − 2.65·18-s − 1.52·19-s − 2.81·20-s + 0.674·21-s + 2.43·22-s − 1.44·23-s − 2.12·24-s + 1.47·25-s − 1.44·26-s + 0.953·27-s + 0.750·28-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 | \( 1 + 18.8T + 128T^{2} \) |
| 3 | \( 1 - 75.2T + 2.18e3T^{2} \) |
| 5 | \( 1 + 439.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 380.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 6.42e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 6.84e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 7.86e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.57e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.41e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.19e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.35e5T + 2.75e10T^{2} \) |
| 41 | \( 1 + 5.37e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.90e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.46e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.05e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 3.07e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.77e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.50e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.10e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 8.75e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.05e3T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.62e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.63e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.79e6T + 8.07e13T^{2} \) |
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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
−14.78018557503130708921003477554, −13.03524640809796966892203523991, −11.31131341899092753910021434814, −10.20551649680435155992111179414, −8.596765912789629018402809335212, −8.169004916736905143073322059641, −7.40344163273226548874239248417, −3.77490353686285789397189534086, −2.09316024970870914687697896024, 0,
2.09316024970870914687697896024, 3.77490353686285789397189534086, 7.40344163273226548874239248417, 8.169004916736905143073322059641, 8.596765912789629018402809335212, 10.20551649680435155992111179414, 11.31131341899092753910021434814, 13.03524640809796966892203523991, 14.78018557503130708921003477554
