| L(s) = 1 | − 2-s + 3.07·3-s + 4-s + 0.560·5-s − 3.07·6-s − 3.76·7-s − 8-s + 6.48·9-s − 0.560·10-s − 0.797·11-s + 3.07·12-s + 5.39·13-s + 3.76·14-s + 1.72·15-s + 16-s − 3.05·17-s − 6.48·18-s + 1.15·19-s + 0.560·20-s − 11.5·21-s + 0.797·22-s − 2.71·23-s − 3.07·24-s − 4.68·25-s − 5.39·26-s + 10.7·27-s − 3.76·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.77·3-s + 0.5·4-s + 0.250·5-s − 1.25·6-s − 1.42·7-s − 0.353·8-s + 2.16·9-s − 0.177·10-s − 0.240·11-s + 0.888·12-s + 1.49·13-s + 1.00·14-s + 0.445·15-s + 0.250·16-s − 0.740·17-s − 1.52·18-s + 0.263·19-s + 0.125·20-s − 2.52·21-s + 0.170·22-s − 0.567·23-s − 0.628·24-s − 0.937·25-s − 1.05·26-s + 2.06·27-s − 0.711·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.576772313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.576772313\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 - 3.07T + 3T^{2} \) |
| 5 | \( 1 - 0.560T + 5T^{2} \) |
| 7 | \( 1 + 3.76T + 7T^{2} \) |
| 11 | \( 1 + 0.797T + 11T^{2} \) |
| 13 | \( 1 - 5.39T + 13T^{2} \) |
| 17 | \( 1 + 3.05T + 17T^{2} \) |
| 19 | \( 1 - 1.15T + 19T^{2} \) |
| 23 | \( 1 + 2.71T + 23T^{2} \) |
| 29 | \( 1 - 7.80T + 29T^{2} \) |
| 31 | \( 1 + 1.50T + 31T^{2} \) |
| 37 | \( 1 - 6.31T + 37T^{2} \) |
| 41 | \( 1 - 2.79T + 41T^{2} \) |
| 43 | \( 1 - 9.79T + 43T^{2} \) |
| 47 | \( 1 - 2.03T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 6.41T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 3.95T + 73T^{2} \) |
| 79 | \( 1 + 6.00T + 79T^{2} \) |
| 83 | \( 1 - 5.07T + 83T^{2} \) |
| 89 | \( 1 + 5.77T + 89T^{2} \) |
| 97 | \( 1 - 4.27T + 97T^{2} \) |
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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
−8.533483106746150115843536913750, −7.976725004563511070556106402560, −7.17831248035257987802547008078, −6.45005687059510532955772644360, −5.83230230400680476525593219641, −4.16717496997897863285182350917, −3.64423424420729335650034926660, −2.76791078376162954262637127875, −2.21338415795037182129672389585, −0.952421104218454531784721948100,
0.952421104218454531784721948100, 2.21338415795037182129672389585, 2.76791078376162954262637127875, 3.64423424420729335650034926660, 4.16717496997897863285182350917, 5.83230230400680476525593219641, 6.45005687059510532955772644360, 7.17831248035257987802547008078, 7.976725004563511070556106402560, 8.533483106746150115843536913750
