| L(s) = 1 | + 1.64·2-s + 0.710·4-s − 2.77·5-s + 1.91·7-s − 2.12·8-s − 4.56·10-s − 1.66·11-s − 4.67·13-s + 3.15·14-s − 4.91·16-s − 2.81·17-s − 19-s − 1.96·20-s − 2.73·22-s + 8.57·23-s + 2.67·25-s − 7.68·26-s + 1.35·28-s − 3.86·29-s − 6.92·31-s − 3.84·32-s − 4.62·34-s − 5.30·35-s + 5.47·37-s − 1.64·38-s + 5.88·40-s + 5.10·41-s + ⋯ |
| L(s) = 1 | + 1.16·2-s + 0.355·4-s − 1.23·5-s + 0.723·7-s − 0.750·8-s − 1.44·10-s − 0.501·11-s − 1.29·13-s + 0.842·14-s − 1.22·16-s − 0.682·17-s − 0.229·19-s − 0.440·20-s − 0.583·22-s + 1.78·23-s + 0.535·25-s − 1.50·26-s + 0.256·28-s − 0.718·29-s − 1.24·31-s − 0.680·32-s − 0.794·34-s − 0.896·35-s + 0.900·37-s − 0.267·38-s + 0.930·40-s + 0.796·41-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
| \(L(1)\) |
\(\approx\) |
\(1.695903239\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.695903239\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.64T + 2T^{2} \) |
| 5 | \( 1 + 2.77T + 5T^{2} \) |
| 7 | \( 1 - 1.91T + 7T^{2} \) |
| 11 | \( 1 + 1.66T + 11T^{2} \) |
| 13 | \( 1 + 4.67T + 13T^{2} \) |
| 17 | \( 1 + 2.81T + 17T^{2} \) |
| 23 | \( 1 - 8.57T + 23T^{2} \) |
| 29 | \( 1 + 3.86T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 - 5.47T + 37T^{2} \) |
| 41 | \( 1 - 5.10T + 41T^{2} \) |
| 43 | \( 1 - 6.35T + 43T^{2} \) |
| 53 | \( 1 + 8.45T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 + 6.03T + 61T^{2} \) |
| 67 | \( 1 - 3.43T + 67T^{2} \) |
| 71 | \( 1 - 6.08T + 71T^{2} \) |
| 73 | \( 1 - 4.60T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 - 5.08T + 83T^{2} \) |
| 89 | \( 1 - 6.11T + 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
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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
−7.47916656026940841638517788685, −7.35610831262896695219430227079, −6.35215917969086698041557915428, −5.39363534250815140227467768942, −4.89571116607364144461769180758, −4.39896836245236878959549878321, −3.71179402708210355556116697415, −2.89107165938709068203295174157, −2.14754666362030166779619254495, −0.51244340901861540020805322072,
0.51244340901861540020805322072, 2.14754666362030166779619254495, 2.89107165938709068203295174157, 3.71179402708210355556116697415, 4.39896836245236878959549878321, 4.89571116607364144461769180758, 5.39363534250815140227467768942, 6.35215917969086698041557915428, 7.35610831262896695219430227079, 7.47916656026940841638517788685
