| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 4.82·11-s + 0.828·13-s + 16-s − 5.41·17-s + 3.41·19-s − 20-s − 4.82·22-s + 6.82·23-s + 25-s + 0.828·26-s − 0.828·29-s + 2.82·31-s + 32-s − 5.41·34-s + 3.65·37-s + 3.41·38-s − 40-s − 11.0·41-s − 3.17·43-s − 4.82·44-s + 6.82·46-s − 10.8·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 1.45·11-s + 0.229·13-s + 0.250·16-s − 1.31·17-s + 0.783·19-s − 0.223·20-s − 1.02·22-s + 1.42·23-s + 0.200·25-s + 0.162·26-s − 0.153·29-s + 0.508·31-s + 0.176·32-s − 0.928·34-s + 0.601·37-s + 0.553·38-s − 0.158·40-s − 1.72·41-s − 0.483·43-s − 0.727·44-s + 1.00·46-s − 1.57·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 - 0.828T + 13T^{2} \) |
| 17 | \( 1 + 5.41T + 17T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 - 6.82T + 23T^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 - 3.65T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 + 3.17T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 + 9.65T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 6.58T + 73T^{2} \) |
| 79 | \( 1 + 1.17T + 79T^{2} \) |
| 83 | \( 1 + 6.24T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−7.88476916109861610507008198953, −7.20057166622237754091645406942, −6.56151052418559675105487493157, −5.63607198433078333643610145539, −4.89786826831927841320854713220, −4.42505851695255334518408492294, −3.20105120032116783877464514057, −2.79837456880461495669087046074, −1.55763784183386075095443219975, 0,
1.55763784183386075095443219975, 2.79837456880461495669087046074, 3.20105120032116783877464514057, 4.42505851695255334518408492294, 4.89786826831927841320854713220, 5.63607198433078333643610145539, 6.56151052418559675105487493157, 7.20057166622237754091645406942, 7.88476916109861610507008198953
