| L(s) = 1 | − 2-s + 4-s − 0.484·5-s − 7-s − 8-s + 0.484·10-s − 6.24·11-s − 1.12·13-s + 14-s + 16-s − 3.60·17-s + 4.64·19-s − 0.484·20-s + 6.24·22-s + 23-s − 4.76·25-s + 1.12·26-s − 28-s + 1.76·29-s − 1.60·31-s − 32-s + 3.60·34-s + 0.484·35-s − 3.76·37-s − 4.64·38-s + 0.484·40-s + 2.73·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.216·5-s − 0.377·7-s − 0.353·8-s + 0.153·10-s − 1.88·11-s − 0.311·13-s + 0.267·14-s + 0.250·16-s − 0.875·17-s + 1.06·19-s − 0.108·20-s + 1.33·22-s + 0.208·23-s − 0.952·25-s + 0.220·26-s − 0.188·28-s + 0.327·29-s − 0.289·31-s − 0.176·32-s + 0.619·34-s + 0.0819·35-s − 0.618·37-s − 0.752·38-s + 0.0766·40-s + 0.427·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7152105975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7152105975\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 0.484T + 5T^{2} \) |
| 11 | \( 1 + 6.24T + 11T^{2} \) |
| 13 | \( 1 + 1.12T + 13T^{2} \) |
| 17 | \( 1 + 3.60T + 17T^{2} \) |
| 19 | \( 1 - 4.64T + 19T^{2} \) |
| 29 | \( 1 - 1.76T + 29T^{2} \) |
| 31 | \( 1 + 1.60T + 31T^{2} \) |
| 37 | \( 1 + 3.76T + 37T^{2} \) |
| 41 | \( 1 - 2.73T + 41T^{2} \) |
| 43 | \( 1 - 6.79T + 43T^{2} \) |
| 47 | \( 1 + 6.15T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 9.21T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 8.31T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 4.24T + 79T^{2} \) |
| 83 | \( 1 + 3.67T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 - 5.12T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.675605657820078765780426376615, −8.024020066526361703933500731093, −7.39730013880484237269451133064, −6.75058096742334625765521149839, −5.63930158470210406056951416215, −5.10582043289920942927495970885, −3.89603570374756048327894118556, −2.84140474575693714984031827231, −2.16260930751603845665469339359, −0.54588673487748155304859310397,
0.54588673487748155304859310397, 2.16260930751603845665469339359, 2.84140474575693714984031827231, 3.89603570374756048327894118556, 5.10582043289920942927495970885, 5.63930158470210406056951416215, 6.75058096742334625765521149839, 7.39730013880484237269451133064, 8.024020066526361703933500731093, 8.675605657820078765780426376615
