| L(s) = 1 | − 2.49·2-s + 3-s + 4.24·4-s + 0.613·5-s − 2.49·6-s − 7-s − 5.61·8-s + 9-s − 1.53·10-s − 2.20·11-s + 4.24·12-s − 6.08·13-s + 2.49·14-s + 0.613·15-s + 5.54·16-s − 4.83·17-s − 2.49·18-s − 2.02·19-s + 2.60·20-s − 21-s + 5.50·22-s + 3.36·23-s − 5.61·24-s − 4.62·25-s + 15.2·26-s + 27-s − 4.24·28-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 0.577·3-s + 2.12·4-s + 0.274·5-s − 1.02·6-s − 0.377·7-s − 1.98·8-s + 0.333·9-s − 0.484·10-s − 0.664·11-s + 1.22·12-s − 1.68·13-s + 0.668·14-s + 0.158·15-s + 1.38·16-s − 1.17·17-s − 0.589·18-s − 0.464·19-s + 0.582·20-s − 0.218·21-s + 1.17·22-s + 0.701·23-s − 1.14·24-s − 0.924·25-s + 2.98·26-s + 0.192·27-s − 0.802·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5552304920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5552304920\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 2.49T + 2T^{2} \) |
| 5 | \( 1 - 0.613T + 5T^{2} \) |
| 11 | \( 1 + 2.20T + 11T^{2} \) |
| 13 | \( 1 + 6.08T + 13T^{2} \) |
| 17 | \( 1 + 4.83T + 17T^{2} \) |
| 19 | \( 1 + 2.02T + 19T^{2} \) |
| 23 | \( 1 - 3.36T + 23T^{2} \) |
| 29 | \( 1 - 2.93T + 29T^{2} \) |
| 31 | \( 1 - 8.74T + 31T^{2} \) |
| 37 | \( 1 - 1.17T + 37T^{2} \) |
| 41 | \( 1 + 3.39T + 41T^{2} \) |
| 43 | \( 1 - 1.86T + 43T^{2} \) |
| 47 | \( 1 - 4.86T + 47T^{2} \) |
| 53 | \( 1 + 0.596T + 53T^{2} \) |
| 59 | \( 1 + 3.43T + 59T^{2} \) |
| 61 | \( 1 + 9.06T + 61T^{2} \) |
| 67 | \( 1 + 4.26T + 67T^{2} \) |
| 71 | \( 1 - 0.0483T + 71T^{2} \) |
| 73 | \( 1 - 4.78T + 73T^{2} \) |
| 79 | \( 1 + 8.43T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 5.56T + 89T^{2} \) |
| 97 | \( 1 - 0.138T + 97T^{2} \) |
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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
−7.912356820149475917891551195495, −7.42016955103539853983436115742, −6.74817283542794079351270412026, −6.20236258807471561513889402117, −5.03794145415813667522793082814, −4.28766819206300355578002737655, −2.81162912708916987680304920635, −2.55960569660070416372094558379, −1.73086339243029615767424609395, −0.44581288586431210675700035851,
0.44581288586431210675700035851, 1.73086339243029615767424609395, 2.55960569660070416372094558379, 2.81162912708916987680304920635, 4.28766819206300355578002737655, 5.03794145415813667522793082814, 6.20236258807471561513889402117, 6.74817283542794079351270412026, 7.42016955103539853983436115742, 7.912356820149475917891551195495
