L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 2·7-s − 8-s + 9-s − 2.64·11-s + 12-s − 4.24·13-s − 2·14-s + 16-s + 6.24·17-s − 18-s + 3.67·19-s + 2·21-s + 2.64·22-s + 23-s − 24-s + 4.24·26-s + 27-s + 2·28-s − 2·29-s + 2.96·31-s − 32-s − 2.64·33-s − 6.24·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 0.333·9-s − 0.795·11-s + 0.288·12-s − 1.17·13-s − 0.534·14-s + 0.250·16-s + 1.51·17-s − 0.235·18-s + 0.842·19-s + 0.436·21-s + 0.562·22-s + 0.208·23-s − 0.204·24-s + 0.833·26-s + 0.192·27-s + 0.377·28-s − 0.371·29-s + 0.533·31-s − 0.176·32-s − 0.459·33-s − 1.07·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.773310903\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.773310903\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 2.64T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 - 6.24T + 17T^{2} \) |
| 19 | \( 1 - 3.67T + 19T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 2.96T + 31T^{2} \) |
| 37 | \( 1 - 3.35T + 37T^{2} \) |
| 41 | \( 1 + 6.24T + 41T^{2} \) |
| 43 | \( 1 + 0.640T + 43T^{2} \) |
| 47 | \( 1 + 4.24T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 8.31T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 - 7.52T + 71T^{2} \) |
| 73 | \( 1 - 9.03T + 73T^{2} \) |
| 79 | \( 1 - 9.52T + 79T^{2} \) |
| 83 | \( 1 - 17.2T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
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
−8.373814375641656500747950741012, −7.913789312162005229255212638700, −7.46940306301753201770865417469, −6.64247774371404850271521038839, −5.33327935415220523420468243913, −5.04854193290841874508224536175, −3.69868166472035592313180982318, −2.81864909970891016864962473196, −2.01333667428818886241839024545, −0.866405691358436335191593652213,
0.866405691358436335191593652213, 2.01333667428818886241839024545, 2.81864909970891016864962473196, 3.69868166472035592313180982318, 5.04854193290841874508224536175, 5.33327935415220523420468243913, 6.64247774371404850271521038839, 7.46940306301753201770865417469, 7.913789312162005229255212638700, 8.373814375641656500747950741012