| L(s) = 1 | + 2.70·2-s − 3-s + 5.30·4-s − 5-s − 2.70·6-s + 8.94·8-s + 9-s − 2.70·10-s − 11-s − 5.30·12-s − 1.39·13-s + 15-s + 13.5·16-s − 4.30·17-s + 2.70·18-s + 2.30·19-s − 5.30·20-s − 2.70·22-s + 7.28·23-s − 8.94·24-s + 25-s − 3.77·26-s − 27-s + 1.27·29-s + 2.70·30-s + 6.85·31-s + 18.7·32-s + ⋯ |
| L(s) = 1 | + 1.91·2-s − 0.577·3-s + 2.65·4-s − 0.447·5-s − 1.10·6-s + 3.16·8-s + 0.333·9-s − 0.854·10-s − 0.301·11-s − 1.53·12-s − 0.387·13-s + 0.258·15-s + 3.38·16-s − 1.04·17-s + 0.637·18-s + 0.529·19-s − 1.18·20-s − 0.576·22-s + 1.51·23-s − 1.82·24-s + 0.200·25-s − 0.739·26-s − 0.192·27-s + 0.237·29-s + 0.493·30-s + 1.23·31-s + 3.31·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.936249864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.936249864\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.70T + 2T^{2} \) |
| 13 | \( 1 + 1.39T + 13T^{2} \) |
| 17 | \( 1 + 4.30T + 17T^{2} \) |
| 19 | \( 1 - 2.30T + 19T^{2} \) |
| 23 | \( 1 - 7.28T + 23T^{2} \) |
| 29 | \( 1 - 1.27T + 29T^{2} \) |
| 31 | \( 1 - 6.85T + 31T^{2} \) |
| 37 | \( 1 + 7.87T + 37T^{2} \) |
| 41 | \( 1 - 8.01T + 41T^{2} \) |
| 43 | \( 1 - 0.296T + 43T^{2} \) |
| 47 | \( 1 - 1.16T + 47T^{2} \) |
| 53 | \( 1 - 9.57T + 53T^{2} \) |
| 59 | \( 1 + 1.33T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 0.418T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 - 3.16T + 79T^{2} \) |
| 83 | \( 1 - 5.51T + 83T^{2} \) |
| 89 | \( 1 - 5.33T + 89T^{2} \) |
| 97 | \( 1 + 9.20T + 97T^{2} \) |
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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
−7.31384742925895388955789814035, −6.93925332654230836493616051423, −6.32790809862592581591054318049, −5.49147748099077033272146382249, −4.97206989037764320757155991802, −4.45409598930836659746939912440, −3.70729960937273039158460071561, −2.88261178182334658262970726422, −2.21281589852067980577771269542, −0.936351401437806724166792345420,
0.936351401437806724166792345420, 2.21281589852067980577771269542, 2.88261178182334658262970726422, 3.70729960937273039158460071561, 4.45409598930836659746939912440, 4.97206989037764320757155991802, 5.49147748099077033272146382249, 6.32790809862592581591054318049, 6.93925332654230836493616051423, 7.31384742925895388955789814035
