| L(s) = 1 | + 4.82·2-s + 15.2·4-s + 17.0·5-s − 0.894·7-s + 34.8·8-s + 82.0·10-s + 12.8·11-s − 55.2·13-s − 4.31·14-s + 46.2·16-s − 45.7·17-s + 57.6·19-s + 259.·20-s + 61.8·22-s + 161.·23-s + 164.·25-s − 266.·26-s − 13.6·28-s + 278.·29-s − 207.·31-s − 56.1·32-s − 220.·34-s − 15.2·35-s + 170.·37-s + 278.·38-s + 593.·40-s + 253.·41-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.90·4-s + 1.52·5-s − 0.0482·7-s + 1.54·8-s + 2.59·10-s + 0.351·11-s − 1.17·13-s − 0.0822·14-s + 0.722·16-s − 0.652·17-s + 0.696·19-s + 2.90·20-s + 0.599·22-s + 1.46·23-s + 1.31·25-s − 2.01·26-s − 0.0919·28-s + 1.78·29-s − 1.20·31-s − 0.310·32-s − 1.11·34-s − 0.0735·35-s + 0.756·37-s + 1.18·38-s + 2.34·40-s + 0.963·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.363349326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.363349326\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 - 4.82T + 8T^{2} \) |
| 5 | \( 1 - 17.0T + 125T^{2} \) |
| 7 | \( 1 + 0.894T + 343T^{2} \) |
| 11 | \( 1 - 12.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 55.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 45.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 57.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 278.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 207.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 170.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 253.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 401.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 339.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 623.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 588.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 397.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 765.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 274.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.18e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 650.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 526.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.21e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.60e3T + 9.12e5T^{2} \) |
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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
−9.014883523827764715003529085944, −7.48344033716028317697060839548, −6.80431267575864074968079650027, −6.16541059097590524769253982792, −5.36603668646724817455069012261, −4.91141083682265456164221635667, −3.98402607329278811125633361458, −2.67740866260769056119044815279, −2.44554149372838682616644532720, −1.13663035494224112481787012993,
1.13663035494224112481787012993, 2.44554149372838682616644532720, 2.67740866260769056119044815279, 3.98402607329278811125633361458, 4.91141083682265456164221635667, 5.36603668646724817455069012261, 6.16541059097590524769253982792, 6.80431267575864074968079650027, 7.48344033716028317697060839548, 9.014883523827764715003529085944
