| L(s) = 1 | − 1.85·2-s + 3-s + 1.43·4-s − 1.85·6-s + 2.79·7-s + 1.05·8-s + 9-s + 0.110·11-s + 1.43·12-s − 6.18·13-s − 5.18·14-s − 4.81·16-s − 4.27·17-s − 1.85·18-s − 0.326·19-s + 2.79·21-s − 0.205·22-s + 5.33·23-s + 1.05·24-s + 11.4·26-s + 27-s + 4.00·28-s − 8.69·29-s − 5.01·31-s + 6.81·32-s + 0.110·33-s + 7.91·34-s + ⋯ |
| L(s) = 1 | − 1.30·2-s + 0.577·3-s + 0.715·4-s − 0.756·6-s + 1.05·7-s + 0.372·8-s + 0.333·9-s + 0.0334·11-s + 0.413·12-s − 1.71·13-s − 1.38·14-s − 1.20·16-s − 1.03·17-s − 0.436·18-s − 0.0747·19-s + 0.610·21-s − 0.0437·22-s + 1.11·23-s + 0.214·24-s + 2.24·26-s + 0.192·27-s + 0.757·28-s − 1.61·29-s − 0.900·31-s + 1.20·32-s + 0.0192·33-s + 1.35·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.084318852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.084318852\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 1.85T + 2T^{2} \) |
| 7 | \( 1 - 2.79T + 7T^{2} \) |
| 11 | \( 1 - 0.110T + 11T^{2} \) |
| 13 | \( 1 + 6.18T + 13T^{2} \) |
| 17 | \( 1 + 4.27T + 17T^{2} \) |
| 19 | \( 1 + 0.326T + 19T^{2} \) |
| 23 | \( 1 - 5.33T + 23T^{2} \) |
| 29 | \( 1 + 8.69T + 29T^{2} \) |
| 31 | \( 1 + 5.01T + 31T^{2} \) |
| 37 | \( 1 - 1.49T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 8.23T + 43T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 - 7.90T + 59T^{2} \) |
| 61 | \( 1 - 9.66T + 61T^{2} \) |
| 67 | \( 1 - 2.31T + 67T^{2} \) |
| 71 | \( 1 + 4.49T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 17.2T + 83T^{2} \) |
| 89 | \( 1 - 5.04T + 89T^{2} \) |
| 97 | \( 1 + 7.54T + 97T^{2} \) |
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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
−8.741123911357162381483547598732, −7.81845474028381785639977490004, −7.43033048679395532180999785485, −6.89803666314292080873125110666, −5.46412314150406754003335287229, −4.71780331263901981168518459416, −3.98069719015441333337534029737, −2.42232150909866044722657428713, −2.02875866431245093551470347744, −0.72263452177008680505011061010,
0.72263452177008680505011061010, 2.02875866431245093551470347744, 2.42232150909866044722657428713, 3.98069719015441333337534029737, 4.71780331263901981168518459416, 5.46412314150406754003335287229, 6.89803666314292080873125110666, 7.43033048679395532180999785485, 7.81845474028381785639977490004, 8.741123911357162381483547598732
