| L(s) = 1 | − 2-s − 2.54·3-s + 4-s − 4.40·5-s + 2.54·6-s − 2.39·7-s − 8-s + 3.47·9-s + 4.40·10-s − 3.64·11-s − 2.54·12-s + 5.13·13-s + 2.39·14-s + 11.2·15-s + 16-s + 2.22·17-s − 3.47·18-s + 19-s − 4.40·20-s + 6.08·21-s + 3.64·22-s + 3.94·23-s + 2.54·24-s + 14.4·25-s − 5.13·26-s − 1.20·27-s − 2.39·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.46·3-s + 0.5·4-s − 1.97·5-s + 1.03·6-s − 0.903·7-s − 0.353·8-s + 1.15·9-s + 1.39·10-s − 1.09·11-s − 0.734·12-s + 1.42·13-s + 0.638·14-s + 2.89·15-s + 0.250·16-s + 0.539·17-s − 0.818·18-s + 0.229·19-s − 0.985·20-s + 1.32·21-s + 0.777·22-s + 0.822·23-s + 0.519·24-s + 2.88·25-s − 1.00·26-s − 0.231·27-s − 0.451·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4165165641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4165165641\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 2.54T + 3T^{2} \) |
| 5 | \( 1 + 4.40T + 5T^{2} \) |
| 7 | \( 1 + 2.39T + 7T^{2} \) |
| 11 | \( 1 + 3.64T + 11T^{2} \) |
| 13 | \( 1 - 5.13T + 13T^{2} \) |
| 17 | \( 1 - 2.22T + 17T^{2} \) |
| 23 | \( 1 - 3.94T + 23T^{2} \) |
| 29 | \( 1 - 4.94T + 29T^{2} \) |
| 31 | \( 1 - 9.26T + 31T^{2} \) |
| 37 | \( 1 + 3.48T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 9.31T + 43T^{2} \) |
| 47 | \( 1 - 7.12T + 47T^{2} \) |
| 53 | \( 1 + 8.00T + 53T^{2} \) |
| 59 | \( 1 + 2.21T + 59T^{2} \) |
| 61 | \( 1 + 7.43T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 + 6.23T + 71T^{2} \) |
| 73 | \( 1 - 4.65T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 - 9.55T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 2.60T + 97T^{2} \) |
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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
−7.956537211262009651866713511755, −7.06343760588656492244590521124, −6.57739284725705818320950714674, −5.94852189435164663094652919394, −5.03425042387154631466267379940, −4.40656196286762554083825945639, −3.39058401380068695030455086139, −2.94597682626761351369294578195, −0.996715753293493910989874244343, −0.51513453261315875315240330543,
0.51513453261315875315240330543, 0.996715753293493910989874244343, 2.94597682626761351369294578195, 3.39058401380068695030455086139, 4.40656196286762554083825945639, 5.03425042387154631466267379940, 5.94852189435164663094652919394, 6.57739284725705818320950714674, 7.06343760588656492244590521124, 7.956537211262009651866713511755
