| L(s) = 1 | − 2-s − 0.767·3-s + 4-s + 2.24·5-s + 0.767·6-s + 7-s − 8-s − 2.41·9-s − 2.24·10-s − 4.92·11-s − 0.767·12-s − 2.23·13-s − 14-s − 1.71·15-s + 16-s + 2.78·17-s + 2.41·18-s + 2.24·20-s − 0.767·21-s + 4.92·22-s + 7.55·23-s + 0.767·24-s + 0.0213·25-s + 2.23·26-s + 4.15·27-s + 28-s − 4.83·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.442·3-s + 0.5·4-s + 1.00·5-s + 0.313·6-s + 0.377·7-s − 0.353·8-s − 0.803·9-s − 0.708·10-s − 1.48·11-s − 0.221·12-s − 0.618·13-s − 0.267·14-s − 0.443·15-s + 0.250·16-s + 0.674·17-s + 0.568·18-s + 0.501·20-s − 0.167·21-s + 1.04·22-s + 1.57·23-s + 0.156·24-s + 0.00426·25-s + 0.437·26-s + 0.798·27-s + 0.188·28-s − 0.896·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.006890114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.006890114\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 0.767T + 3T^{2} \) |
| 5 | \( 1 - 2.24T + 5T^{2} \) |
| 11 | \( 1 + 4.92T + 11T^{2} \) |
| 13 | \( 1 + 2.23T + 13T^{2} \) |
| 17 | \( 1 - 2.78T + 17T^{2} \) |
| 23 | \( 1 - 7.55T + 23T^{2} \) |
| 29 | \( 1 + 4.83T + 29T^{2} \) |
| 31 | \( 1 + 9.15T + 31T^{2} \) |
| 37 | \( 1 - 0.687T + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 + 9.26T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 2.50T + 59T^{2} \) |
| 61 | \( 1 - 8.72T + 61T^{2} \) |
| 67 | \( 1 - 2.20T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 5.63T + 73T^{2} \) |
| 79 | \( 1 - 3.82T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 + 2.39T + 89T^{2} \) |
| 97 | \( 1 - 5.96T + 97T^{2} \) |
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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
−8.300236651682954001322313512064, −7.48294767536824794145790504867, −6.99100915432078275553755639825, −5.89249091385732832434863010808, −5.36647120092168872297322029040, −5.06754002675408223325242550750, −3.46252186845877877935096313614, −2.55912436201938758068374287902, −1.92122444766380170644857632051, −0.59914641817009420182873186873,
0.59914641817009420182873186873, 1.92122444766380170644857632051, 2.55912436201938758068374287902, 3.46252186845877877935096313614, 5.06754002675408223325242550750, 5.36647120092168872297322029040, 5.89249091385732832434863010808, 6.99100915432078275553755639825, 7.48294767536824794145790504867, 8.300236651682954001322313512064
