| L(s) = 1 | − 1.61·2-s + 0.618·4-s − 5-s − 0.236·7-s + 2.23·8-s + 1.61·10-s − 4.23·11-s + 0.381·14-s − 4.85·16-s − 5.47·17-s − 0.236·19-s − 0.618·20-s + 6.85·22-s − 8.23·23-s + 25-s − 0.145·28-s − 1.47·29-s + 3.38·32-s + 8.85·34-s + 0.236·35-s + 3·37-s + 0.381·38-s − 2.23·40-s + 5.94·41-s − 1.76·43-s − 2.61·44-s + 13.3·46-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.309·4-s − 0.447·5-s − 0.0892·7-s + 0.790·8-s + 0.511·10-s − 1.27·11-s + 0.102·14-s − 1.21·16-s − 1.32·17-s − 0.0541·19-s − 0.138·20-s + 1.46·22-s − 1.71·23-s + 0.200·25-s − 0.0275·28-s − 0.273·29-s + 0.597·32-s + 1.51·34-s + 0.0399·35-s + 0.493·37-s + 0.0619·38-s − 0.353·40-s + 0.928·41-s − 0.268·43-s − 0.394·44-s + 1.96·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1668229512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1668229512\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 7 | \( 1 + 0.236T + 7T^{2} \) |
| 11 | \( 1 + 4.23T + 11T^{2} \) |
| 17 | \( 1 + 5.47T + 17T^{2} \) |
| 19 | \( 1 + 0.236T + 19T^{2} \) |
| 23 | \( 1 + 8.23T + 23T^{2} \) |
| 29 | \( 1 + 1.47T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 - 5.94T + 41T^{2} \) |
| 43 | \( 1 + 1.76T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 - 1.76T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 8.94T + 83T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 - 5.47T + 97T^{2} \) |
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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
−7.913088806156248385426142568803, −7.61307963128887356805799848546, −6.63452530478965462220040209748, −6.00567527137617677149679835212, −4.84729009224135374961227285619, −4.48979748736442771743272636195, −3.45744916761800363479144648807, −2.41594350724771438016997998885, −1.66949457576490221485770993179, −0.23750434799532631638834351412,
0.23750434799532631638834351412, 1.66949457576490221485770993179, 2.41594350724771438016997998885, 3.45744916761800363479144648807, 4.48979748736442771743272636195, 4.84729009224135374961227285619, 6.00567527137617677149679835212, 6.63452530478965462220040209748, 7.61307963128887356805799848546, 7.913088806156248385426142568803
