| L(s) = 1 | + 1.19·2-s − 0.576·4-s − 1.08·5-s − 1.30·7-s − 3.07·8-s − 1.29·10-s + 11-s + 2.53·13-s − 1.56·14-s − 2.51·16-s + 5.43·17-s + 19-s + 0.623·20-s + 1.19·22-s − 3.87·23-s − 3.82·25-s + 3.03·26-s + 0.754·28-s + 2.41·29-s + 3.03·31-s + 3.14·32-s + 6.48·34-s + 1.41·35-s + 6.85·37-s + 1.19·38-s + 3.32·40-s + 6.11·41-s + ⋯ |
| L(s) = 1 | + 0.843·2-s − 0.288·4-s − 0.484·5-s − 0.494·7-s − 1.08·8-s − 0.408·10-s + 0.301·11-s + 0.704·13-s − 0.417·14-s − 0.628·16-s + 1.31·17-s + 0.229·19-s + 0.139·20-s + 0.254·22-s − 0.807·23-s − 0.765·25-s + 0.594·26-s + 0.142·28-s + 0.448·29-s + 0.545·31-s + 0.556·32-s + 1.11·34-s + 0.239·35-s + 1.12·37-s + 0.193·38-s + 0.526·40-s + 0.954·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.918778482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.918778482\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.19T + 2T^{2} \) |
| 5 | \( 1 + 1.08T + 5T^{2} \) |
| 7 | \( 1 + 1.30T + 7T^{2} \) |
| 13 | \( 1 - 2.53T + 13T^{2} \) |
| 17 | \( 1 - 5.43T + 17T^{2} \) |
| 23 | \( 1 + 3.87T + 23T^{2} \) |
| 29 | \( 1 - 2.41T + 29T^{2} \) |
| 31 | \( 1 - 3.03T + 31T^{2} \) |
| 37 | \( 1 - 6.85T + 37T^{2} \) |
| 41 | \( 1 - 6.11T + 41T^{2} \) |
| 43 | \( 1 + 2.95T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 - 0.992T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 + 5.82T + 61T^{2} \) |
| 67 | \( 1 - 8.79T + 67T^{2} \) |
| 71 | \( 1 - 2.44T + 71T^{2} \) |
| 73 | \( 1 - 5.84T + 73T^{2} \) |
| 79 | \( 1 - 17.0T + 79T^{2} \) |
| 83 | \( 1 - 7.01T + 83T^{2} \) |
| 89 | \( 1 - 9.13T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
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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
−9.367873363522709536639391842119, −8.306008344928859887793128379878, −7.77425248603725662767184238162, −6.56274010481225011859369230947, −5.93020880892929800496132444305, −5.17723017347159556447726811935, −4.01553981921566074711944945812, −3.69749194964756242402871313388, −2.62949448611511245027306210203, −0.837710298294896467161867034901,
0.837710298294896467161867034901, 2.62949448611511245027306210203, 3.69749194964756242402871313388, 4.01553981921566074711944945812, 5.17723017347159556447726811935, 5.93020880892929800496132444305, 6.56274010481225011859369230947, 7.77425248603725662767184238162, 8.306008344928859887793128379878, 9.367873363522709536639391842119
