| L(s) = 1 | + 0.743·2-s − 1.44·4-s − 0.245·5-s + 1.64·7-s − 2.56·8-s − 0.182·10-s + 0.146·11-s + 3.28·13-s + 1.22·14-s + 0.987·16-s − 3.43·17-s − 19-s + 0.355·20-s + 0.108·22-s + 1.56·23-s − 4.93·25-s + 2.44·26-s − 2.37·28-s − 5.61·29-s + 7.94·31-s + 5.86·32-s − 2.55·34-s − 0.403·35-s − 0.199·37-s − 0.743·38-s + 0.629·40-s − 1.51·41-s + ⋯ |
| L(s) = 1 | + 0.525·2-s − 0.723·4-s − 0.109·5-s + 0.620·7-s − 0.906·8-s − 0.0577·10-s + 0.0441·11-s + 0.910·13-s + 0.326·14-s + 0.246·16-s − 0.834·17-s − 0.229·19-s + 0.0794·20-s + 0.0232·22-s + 0.325·23-s − 0.987·25-s + 0.478·26-s − 0.448·28-s − 1.04·29-s + 1.42·31-s + 1.03·32-s − 0.438·34-s − 0.0681·35-s − 0.0328·37-s − 0.120·38-s + 0.0995·40-s − 0.236·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 0.743T + 2T^{2} \) |
| 5 | \( 1 + 0.245T + 5T^{2} \) |
| 7 | \( 1 - 1.64T + 7T^{2} \) |
| 11 | \( 1 - 0.146T + 11T^{2} \) |
| 13 | \( 1 - 3.28T + 13T^{2} \) |
| 17 | \( 1 + 3.43T + 17T^{2} \) |
| 23 | \( 1 - 1.56T + 23T^{2} \) |
| 29 | \( 1 + 5.61T + 29T^{2} \) |
| 31 | \( 1 - 7.94T + 31T^{2} \) |
| 37 | \( 1 + 0.199T + 37T^{2} \) |
| 41 | \( 1 + 1.51T + 41T^{2} \) |
| 43 | \( 1 + 4.83T + 43T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 5.04T + 59T^{2} \) |
| 61 | \( 1 + 7.40T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 + 4.70T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 5.89T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 - 6.50T + 97T^{2} \) |
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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
−7.59287271547654393735604082207, −6.54924439904997994966443469491, −6.08921364733130928884218431988, −5.21179849682681472678082119165, −4.67950456054343301354942320052, −3.95671578999148137334736143524, −3.37971962313370138473729786200, −2.29333372393883687122569003191, −1.26268742726166232005153922451, 0,
1.26268742726166232005153922451, 2.29333372393883687122569003191, 3.37971962313370138473729786200, 3.95671578999148137334736143524, 4.67950456054343301354942320052, 5.21179849682681472678082119165, 6.08921364733130928884218431988, 6.54924439904997994966443469491, 7.59287271547654393735604082207
