| L(s) = 1 | + 2-s + 0.828·3-s + 4-s + 5-s + 0.828·6-s − 0.496·7-s + 8-s − 2.31·9-s + 10-s − 4.02·11-s + 0.828·12-s + 13-s − 0.496·14-s + 0.828·15-s + 16-s − 7.99·17-s − 2.31·18-s + 2.08·19-s + 20-s − 0.411·21-s − 4.02·22-s − 3.78·23-s + 0.828·24-s + 25-s + 26-s − 4.40·27-s − 0.496·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.478·3-s + 0.5·4-s + 0.447·5-s + 0.338·6-s − 0.187·7-s + 0.353·8-s − 0.771·9-s + 0.316·10-s − 1.21·11-s + 0.239·12-s + 0.277·13-s − 0.132·14-s + 0.213·15-s + 0.250·16-s − 1.93·17-s − 0.545·18-s + 0.478·19-s + 0.223·20-s − 0.0897·21-s − 0.857·22-s − 0.789·23-s + 0.169·24-s + 0.200·25-s + 0.196·26-s − 0.847·27-s − 0.0937·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 - 0.828T + 3T^{2} \) |
| 7 | \( 1 + 0.496T + 7T^{2} \) |
| 11 | \( 1 + 4.02T + 11T^{2} \) |
| 17 | \( 1 + 7.99T + 17T^{2} \) |
| 19 | \( 1 - 2.08T + 19T^{2} \) |
| 23 | \( 1 + 3.78T + 23T^{2} \) |
| 29 | \( 1 + 1.95T + 29T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 - 9.03T + 41T^{2} \) |
| 43 | \( 1 - 8.34T + 43T^{2} \) |
| 47 | \( 1 + 5.66T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 - 2.20T + 59T^{2} \) |
| 61 | \( 1 - 4.99T + 61T^{2} \) |
| 67 | \( 1 - 3.35T + 67T^{2} \) |
| 71 | \( 1 + 8.59T + 71T^{2} \) |
| 73 | \( 1 - 3.79T + 73T^{2} \) |
| 79 | \( 1 - 7.05T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 + 7.98T + 97T^{2} \) |
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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
−8.093528138794607862914211516943, −7.30982168323651097884259657915, −6.43988610664398681312670695411, −5.80997321717174564940447427189, −5.10440909599278215463290977180, −4.27958323590812254683804057124, −3.29303826032405846163772610993, −2.56971161428728053575873338889, −1.90350291735546057381527400828, 0,
1.90350291735546057381527400828, 2.56971161428728053575873338889, 3.29303826032405846163772610993, 4.27958323590812254683804057124, 5.10440909599278215463290977180, 5.80997321717174564940447427189, 6.43988610664398681312670695411, 7.30982168323651097884259657915, 8.093528138794607862914211516943
