| L(s) = 1 | + 38.1·2-s + 139.·3-s + 939.·4-s − 566.·5-s + 5.29e3·6-s + 4.62e3·7-s + 1.63e4·8-s − 356.·9-s − 2.15e4·10-s + 8.60e4·11-s + 1.30e5·12-s − 1.66e4·13-s + 1.76e5·14-s − 7.87e4·15-s + 1.40e5·16-s − 2.27e5·17-s − 1.35e4·18-s + 4.27e5·19-s − 5.32e5·20-s + 6.43e5·21-s + 3.27e6·22-s − 1.52e6·23-s + 2.26e6·24-s − 1.63e6·25-s − 6.35e5·26-s − 2.78e6·27-s + 4.35e6·28-s + ⋯ |
| L(s) = 1 | + 1.68·2-s + 0.990·3-s + 1.83·4-s − 0.405·5-s + 1.66·6-s + 0.728·7-s + 1.40·8-s − 0.0181·9-s − 0.682·10-s + 1.77·11-s + 1.81·12-s − 0.161·13-s + 1.22·14-s − 0.401·15-s + 0.534·16-s − 0.659·17-s − 0.0305·18-s + 0.753·19-s − 0.744·20-s + 0.722·21-s + 2.98·22-s − 1.13·23-s + 1.39·24-s − 0.835·25-s − 0.272·26-s − 1.00·27-s + 1.33·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(6.635730784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.635730784\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 - 1.87e6T \) |
| good | 2 | \( 1 - 38.1T + 512T^{2} \) |
| 3 | \( 1 - 139.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 566.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 4.62e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 8.60e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.66e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.27e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 4.27e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.52e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 7.81e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.82e5T + 2.64e13T^{2} \) |
| 41 | \( 1 - 1.60e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.66e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.43e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.18e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.33e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 6.71e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.62e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.24e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.48e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.15e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.76e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 8.76e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.66e9T + 7.60e17T^{2} \) |
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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
−14.32924114708945448012286240929, −13.59647836986753964360355967322, −12.04756896658049177346008664297, −11.39022910596487122583157065986, −9.172102663557928624031830481817, −7.68948286328004500220943673309, −6.14835715784391545444629383024, −4.42914555342319701569083808469, −3.47907430838031673731989752511, −1.94741388577567339971367672887,
1.94741388577567339971367672887, 3.47907430838031673731989752511, 4.42914555342319701569083808469, 6.14835715784391545444629383024, 7.68948286328004500220943673309, 9.172102663557928624031830481817, 11.39022910596487122583157065986, 12.04756896658049177346008664297, 13.59647836986753964360355967322, 14.32924114708945448012286240929
