| L(s) = 1 | − 2-s + 4-s − 2·5-s + 7-s − 8-s + 2·10-s + 4.89·11-s + 2.89·13-s − 14-s + 16-s + 17-s + 4.89·19-s − 2·20-s − 4.89·22-s + 4·23-s − 25-s − 2.89·26-s + 28-s − 6.89·29-s − 9.79·31-s − 32-s − 34-s − 2·35-s − 6.89·37-s − 4.89·38-s + 2·40-s + 2·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.894·5-s + 0.377·7-s − 0.353·8-s + 0.632·10-s + 1.47·11-s + 0.804·13-s − 0.267·14-s + 0.250·16-s + 0.242·17-s + 1.12·19-s − 0.447·20-s − 1.04·22-s + 0.834·23-s − 0.200·25-s − 0.568·26-s + 0.188·28-s − 1.28·29-s − 1.75·31-s − 0.176·32-s − 0.171·34-s − 0.338·35-s − 1.13·37-s − 0.794·38-s + 0.316·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.266920705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.266920705\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2T + 5T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 - 2.89T + 13T^{2} \) |
| 19 | \( 1 - 4.89T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 6.89T + 29T^{2} \) |
| 31 | \( 1 + 9.79T + 31T^{2} \) |
| 37 | \( 1 + 6.89T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 0.898T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 5.79T + 67T^{2} \) |
| 71 | \( 1 - 5.79T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 9.79T + 79T^{2} \) |
| 83 | \( 1 - 7.10T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
| 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
−9.147080369599292490919996109888, −8.372541820678954161124406893595, −7.48571867361843685659515974120, −7.10067312471550220789919331329, −6.04805012883477886239974188303, −5.17542264384621679892914186148, −3.81946932482990303785827487700, −3.52370330843342571900505093837, −1.86784481485873700349840901877, −0.866584398421233322807915499967,
0.866584398421233322807915499967, 1.86784481485873700349840901877, 3.52370330843342571900505093837, 3.81946932482990303785827487700, 5.17542264384621679892914186148, 6.04805012883477886239974188303, 7.10067312471550220789919331329, 7.48571867361843685659515974120, 8.372541820678954161124406893595, 9.147080369599292490919996109888
