| L(s) = 1 | + 2-s − 1.74·3-s + 4-s + 1.36·5-s − 1.74·6-s + 0.246·7-s + 8-s + 0.0402·9-s + 1.36·10-s + 0.997·11-s − 1.74·12-s + 1.44·13-s + 0.246·14-s − 2.37·15-s + 16-s + 7.37·17-s + 0.0402·18-s + 0.127·19-s + 1.36·20-s − 0.429·21-s + 0.997·22-s − 23-s − 1.74·24-s − 3.13·25-s + 1.44·26-s + 5.16·27-s + 0.246·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.00·3-s + 0.5·4-s + 0.610·5-s − 0.711·6-s + 0.0930·7-s + 0.353·8-s + 0.0134·9-s + 0.431·10-s + 0.300·11-s − 0.503·12-s + 0.400·13-s + 0.0658·14-s − 0.614·15-s + 0.250·16-s + 1.78·17-s + 0.00948·18-s + 0.0291·19-s + 0.305·20-s − 0.0936·21-s + 0.212·22-s − 0.208·23-s − 0.355·24-s − 0.627·25-s + 0.282·26-s + 0.993·27-s + 0.0465·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.698215891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.698215891\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 + 1.74T + 3T^{2} \) |
| 5 | \( 1 - 1.36T + 5T^{2} \) |
| 7 | \( 1 - 0.246T + 7T^{2} \) |
| 11 | \( 1 - 0.997T + 11T^{2} \) |
| 13 | \( 1 - 1.44T + 13T^{2} \) |
| 17 | \( 1 - 7.37T + 17T^{2} \) |
| 19 | \( 1 - 0.127T + 19T^{2} \) |
| 29 | \( 1 - 3.04T + 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 - 2.76T + 37T^{2} \) |
| 41 | \( 1 - 9.95T + 41T^{2} \) |
| 43 | \( 1 - 8.18T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 - 3.11T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 2.02T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 - 9.72T + 71T^{2} \) |
| 73 | \( 1 + 2.07T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + 0.120T + 83T^{2} \) |
| 89 | \( 1 - 0.217T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−7.85080679519977161436939548523, −7.23660977576375086499001686717, −6.20698165620726988074076737319, −5.93143624267672384894864672660, −5.39346874574761615412883202099, −4.62498842358646307095298971489, −3.71540671224451620678943378720, −2.92157787844975775879603502423, −1.80115419590516532840858201690, −0.847962870319671008243878487357,
0.847962870319671008243878487357, 1.80115419590516532840858201690, 2.92157787844975775879603502423, 3.71540671224451620678943378720, 4.62498842358646307095298971489, 5.39346874574761615412883202099, 5.93143624267672384894864672660, 6.20698165620726988074076737319, 7.23660977576375086499001686717, 7.85080679519977161436939548523
