L(s) = 1 | − 22.0·2-s + 104.·3-s − 24.1·4-s + 861.·5-s − 2.29e3·6-s − 5.59e3·7-s + 1.18e4·8-s − 8.84e3·9-s − 1.90e4·10-s + 1.60e4·11-s − 2.51e3·12-s + 4.68e4·13-s + 1.23e5·14-s + 8.97e4·15-s − 2.49e5·16-s + 4.31e5·17-s + 1.95e5·18-s − 8.70e5·19-s − 2.08e4·20-s − 5.82e5·21-s − 3.54e5·22-s + 1.85e5·23-s + 1.23e6·24-s − 1.21e6·25-s − 1.03e6·26-s − 2.96e6·27-s + 1.35e5·28-s + ⋯ |
L(s) = 1 | − 0.976·2-s + 0.742·3-s − 0.0471·4-s + 0.616·5-s − 0.724·6-s − 0.880·7-s + 1.02·8-s − 0.449·9-s − 0.601·10-s + 0.330·11-s − 0.0350·12-s + 0.454·13-s + 0.859·14-s + 0.457·15-s − 0.950·16-s + 1.25·17-s + 0.438·18-s − 1.53·19-s − 0.0290·20-s − 0.653·21-s − 0.322·22-s + 0.138·23-s + 0.758·24-s − 0.619·25-s − 0.443·26-s − 1.07·27-s + 0.0415·28-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 | \( 1 + 22.0T + 512T^{2} \) |
| 3 | \( 1 - 104.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 861.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 5.59e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 1.60e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 4.68e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 4.31e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.70e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.85e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.35e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.47e6T + 2.64e13T^{2} \) |
| 41 | \( 1 - 1.64e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.31e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.85e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.33e5T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.67e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 4.68e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.68e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 5.13e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.30e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.20e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.31e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.48e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.63e8T + 7.60e17T^{2} \) |
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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
−13.79147493097084215577757447725, −12.81443316382558755043716890052, −10.83253331275906811391851828725, −9.573534603771880349992307589543, −8.941906158935560095831349455347, −7.65483956050823166348880790636, −5.90777032386140944729195667312, −3.63911127656554038524547518384, −1.85189746689486432281318875799, 0,
1.85189746689486432281318875799, 3.63911127656554038524547518384, 5.90777032386140944729195667312, 7.65483956050823166348880790636, 8.941906158935560095831349455347, 9.573534603771880349992307589543, 10.83253331275906811391851828725, 12.81443316382558755043716890052, 13.79147493097084215577757447725