| L(s) = 1 | − 12.5·2-s − 77.5·3-s + 29.8·4-s + 318.·5-s + 974.·6-s − 75.8·7-s + 1.23e3·8-s + 3.83e3·9-s − 3.99e3·10-s − 266.·11-s − 2.31e3·12-s − 8.01e3·13-s + 953.·14-s − 2.46e4·15-s − 1.93e4·16-s + 3.52e4·17-s − 4.81e4·18-s + 9.44e3·19-s + 9.49e3·20-s + 5.88e3·21-s + 3.34e3·22-s − 8.37e4·23-s − 9.56e4·24-s + 2.31e4·25-s + 1.00e5·26-s − 1.27e5·27-s − 2.26e3·28-s + ⋯ |
| L(s) = 1 | − 1.11·2-s − 1.65·3-s + 0.233·4-s + 1.13·5-s + 1.84·6-s − 0.0836·7-s + 0.851·8-s + 1.75·9-s − 1.26·10-s − 0.0603·11-s − 0.386·12-s − 1.01·13-s + 0.0928·14-s − 1.88·15-s − 1.17·16-s + 1.74·17-s − 1.94·18-s + 0.315·19-s + 0.265·20-s + 0.138·21-s + 0.0670·22-s − 1.43·23-s − 1.41·24-s + 0.296·25-s + 1.12·26-s − 1.24·27-s − 0.0194·28-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 | \( 1 + 12.5T + 128T^{2} \) |
| 3 | \( 1 + 77.5T + 2.18e3T^{2} \) |
| 5 | \( 1 - 318.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 75.8T + 8.23e5T^{2} \) |
| 11 | \( 1 + 266.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 8.01e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.52e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 9.44e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.37e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 7.36e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.02e5T + 2.75e10T^{2} \) |
| 41 | \( 1 - 4.78e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.55e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.25e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.59e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 6.13e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 8.22e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.31e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 7.30e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 9.30e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.74e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.81e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.08e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.09e7T + 8.07e13T^{2} \) |
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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
−14.19700876563988607484996801327, −12.74888800970634156906057923357, −11.53937114745481859294940008041, −9.977546762877781455543972113175, −9.899143947578309688104621973816, −7.68510540168877106689956939509, −6.14302205082673173302448977202, −5.00859050087295657159588040977, −1.48211141174816756662057767515, 0,
1.48211141174816756662057767515, 5.00859050087295657159588040977, 6.14302205082673173302448977202, 7.68510540168877106689956939509, 9.899143947578309688104621973816, 9.977546762877781455543972113175, 11.53937114745481859294940008041, 12.74888800970634156906057923357, 14.19700876563988607484996801327
