| L(s) = 1 | − 2-s + 4-s − 2·5-s − 1.56·7-s − 8-s + 2·10-s + 1.56·11-s + 6.68·13-s + 1.56·14-s + 16-s − 3.56·17-s − 3.56·19-s − 2·20-s − 1.56·22-s − 8.68·23-s − 25-s − 6.68·26-s − 1.56·28-s + 1.12·29-s − 9.12·31-s − 32-s + 3.56·34-s + 3.12·35-s − 37-s + 3.56·38-s + 2·40-s − 11.1·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.894·5-s − 0.590·7-s − 0.353·8-s + 0.632·10-s + 0.470·11-s + 1.85·13-s + 0.417·14-s + 0.250·16-s − 0.863·17-s − 0.817·19-s − 0.447·20-s − 0.332·22-s − 1.81·23-s − 0.200·25-s − 1.31·26-s − 0.295·28-s + 0.208·29-s − 1.63·31-s − 0.176·32-s + 0.610·34-s + 0.527·35-s − 0.164·37-s + 0.577·38-s + 0.316·40-s − 1.73·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 13 | \( 1 - 6.68T + 13T^{2} \) |
| 17 | \( 1 + 3.56T + 17T^{2} \) |
| 19 | \( 1 + 3.56T + 19T^{2} \) |
| 23 | \( 1 + 8.68T + 23T^{2} \) |
| 29 | \( 1 - 1.12T + 29T^{2} \) |
| 31 | \( 1 + 9.12T + 31T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 - 1.12T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 1.56T + 53T^{2} \) |
| 59 | \( 1 + 0.876T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 2.24T + 67T^{2} \) |
| 71 | \( 1 + 2.24T + 71T^{2} \) |
| 73 | \( 1 - 3.56T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 2.87T + 97T^{2} \) |
| 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
−10.13584358345770401042953234905, −8.975123713168213773965377021203, −8.523746805209980316711267621929, −7.59496725573606989190371983421, −6.55607500307732686649523299053, −5.93492862049092353964252060622, −4.13297606196451654133338922812, −3.51422167165774756982684200647, −1.81225175702583064467541270675, 0,
1.81225175702583064467541270675, 3.51422167165774756982684200647, 4.13297606196451654133338922812, 5.93492862049092353964252060622, 6.55607500307732686649523299053, 7.59496725573606989190371983421, 8.523746805209980316711267621929, 8.975123713168213773965377021203, 10.13584358345770401042953234905
