| L(s) = 1 | + 0.674·2-s − 256.·3-s − 511.·4-s + 2.35e3·5-s − 173.·6-s + 2.77e3·7-s − 690.·8-s + 4.61e4·9-s + 1.58e3·10-s − 3.12e4·11-s + 1.31e5·12-s + 3.73e4·13-s + 1.86e3·14-s − 6.04e5·15-s + 2.61e5·16-s − 3.56e5·17-s + 3.11e4·18-s − 8.16e5·19-s − 1.20e6·20-s − 7.11e5·21-s − 2.10e4·22-s + 2.46e6·23-s + 1.77e5·24-s + 3.58e6·25-s + 2.51e4·26-s − 6.79e6·27-s − 1.41e6·28-s + ⋯ |
| L(s) = 1 | + 0.0298·2-s − 1.82·3-s − 0.999·4-s + 1.68·5-s − 0.0545·6-s + 0.436·7-s − 0.0595·8-s + 2.34·9-s + 0.0502·10-s − 0.643·11-s + 1.82·12-s + 0.362·13-s + 0.0130·14-s − 3.08·15-s + 0.997·16-s − 1.03·17-s + 0.0699·18-s − 1.43·19-s − 1.68·20-s − 0.798·21-s − 0.0191·22-s + 1.83·23-s + 0.108·24-s + 1.83·25-s + 0.0107·26-s − 2.46·27-s − 0.435·28-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
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 0.674T + 512T^{2} \) |
| 3 | \( 1 + 256.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.35e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 2.77e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.12e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 3.73e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.56e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.16e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.46e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.35e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.75e6T + 2.64e13T^{2} \) |
| 41 | \( 1 + 7.91e5T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.72e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.45e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.04e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 4.75e6T + 8.66e15T^{2} \) |
| 61 | \( 1 + 9.53e6T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.64e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.15e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.11e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.51e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 7.62e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 9.11e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.08e9T + 7.60e17T^{2} \) |
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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
−13.20504868815962021053318985437, −12.95113924339429638188458585238, −11.05017401759320269513298300820, −10.28150951535704882639776190604, −8.989393277572001827208059543001, −6.61721198348336016454512365151, −5.49858910311943779176810781136, −4.74895267673849078214682771861, −1.56301998421231030290602563923, 0,
1.56301998421231030290602563923, 4.74895267673849078214682771861, 5.49858910311943779176810781136, 6.61721198348336016454512365151, 8.989393277572001827208059543001, 10.28150951535704882639776190604, 11.05017401759320269513298300820, 12.95113924339429638188458585238, 13.20504868815962021053318985437
