| L(s) = 1 | + 2-s + 4-s + 3.23·5-s + 0.236·7-s + 8-s + 3.23·10-s − 0.236·11-s + 3.47·13-s + 0.236·14-s + 16-s − 2·17-s + 5.70·19-s + 3.23·20-s − 0.236·22-s + 2.23·23-s + 5.47·25-s + 3.47·26-s + 0.236·28-s + 7.23·29-s − 5.70·31-s + 32-s − 2·34-s + 0.763·35-s + 4.47·37-s + 5.70·38-s + 3.23·40-s + 0.763·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.44·5-s + 0.0892·7-s + 0.353·8-s + 1.02·10-s − 0.0711·11-s + 0.962·13-s + 0.0630·14-s + 0.250·16-s − 0.485·17-s + 1.30·19-s + 0.723·20-s − 0.0503·22-s + 0.466·23-s + 1.09·25-s + 0.680·26-s + 0.0446·28-s + 1.34·29-s − 1.02·31-s + 0.176·32-s − 0.342·34-s + 0.129·35-s + 0.735·37-s + 0.925·38-s + 0.511·40-s + 0.119·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.306153309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.306153309\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 7 | \( 1 - 0.236T + 7T^{2} \) |
| 11 | \( 1 + 0.236T + 11T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 5.70T + 19T^{2} \) |
| 23 | \( 1 - 2.23T + 23T^{2} \) |
| 29 | \( 1 - 7.23T + 29T^{2} \) |
| 31 | \( 1 + 5.70T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 - 0.763T + 41T^{2} \) |
| 43 | \( 1 - 5.76T + 43T^{2} \) |
| 47 | \( 1 - 0.236T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 + 1.47T + 61T^{2} \) |
| 67 | \( 1 + 8.70T + 67T^{2} \) |
| 71 | \( 1 + 5.23T + 71T^{2} \) |
| 73 | \( 1 + 16.4T + 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 + 3.29T + 83T^{2} \) |
| 89 | \( 1 - 5.52T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−7.52238707641174909433118513808, −6.89617520307567993303444170439, −6.02299789004834311167768426206, −5.86813556312406987016970151548, −5.00554676658070648157872920053, −4.39306176035987668325244594483, −3.31719500260229202229326901220, −2.74130882802419904258600557910, −1.79616167342837031141011850442, −1.09083081873677430114579813790,
1.09083081873677430114579813790, 1.79616167342837031141011850442, 2.74130882802419904258600557910, 3.31719500260229202229326901220, 4.39306176035987668325244594483, 5.00554676658070648157872920053, 5.86813556312406987016970151548, 6.02299789004834311167768426206, 6.89617520307567993303444170439, 7.52238707641174909433118513808
