| L(s) = 1 | + 2-s + 4-s − 5-s − 3.12·7-s + 8-s − 10-s + 2·11-s + 4·13-s − 3.12·14-s + 16-s + 3.12·17-s − 19-s − 20-s + 2·22-s + 25-s + 4·26-s − 3.12·28-s − 2·29-s + 9.12·31-s + 32-s + 3.12·34-s + 3.12·35-s − 38-s − 40-s − 5.12·41-s + 10.2·43-s + 2·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s − 1.18·7-s + 0.353·8-s − 0.316·10-s + 0.603·11-s + 1.10·13-s − 0.834·14-s + 0.250·16-s + 0.757·17-s − 0.229·19-s − 0.223·20-s + 0.426·22-s + 0.200·25-s + 0.784·26-s − 0.590·28-s − 0.371·29-s + 1.63·31-s + 0.176·32-s + 0.535·34-s + 0.527·35-s − 0.162·38-s − 0.158·40-s − 0.800·41-s + 1.56·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.409643911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.409643911\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 3.12T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 3.12T + 17T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 9.12T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 5.12T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 + 3.12T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 - 6.24T + 67T^{2} \) |
| 71 | \( 1 - 6.24T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 9.12T + 79T^{2} \) |
| 83 | \( 1 - 6.87T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−9.377066340388711656351793919491, −8.505170069146041669368657992845, −7.64656640993540486062457393481, −6.63283353310251304038819269910, −6.23941617932906518089671523408, −5.29048096807453615851225433327, −4.05195296127906493418242332683, −3.59658785251440597450313424611, −2.61640077576820863844940973511, −1.00477982265802441032029886721,
1.00477982265802441032029886721, 2.61640077576820863844940973511, 3.59658785251440597450313424611, 4.05195296127906493418242332683, 5.29048096807453615851225433327, 6.23941617932906518089671523408, 6.63283353310251304038819269910, 7.64656640993540486062457393481, 8.505170069146041669368657992845, 9.377066340388711656351793919491
