| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s + 9-s + 11-s + 2·12-s − 6·13-s − 4·16-s − 7·17-s + 2·18-s + 5·19-s + 2·22-s + 23-s − 12·26-s + 27-s − 5·29-s + 8·31-s − 8·32-s + 33-s − 14·34-s + 2·36-s + 2·37-s + 10·38-s − 6·39-s − 12·41-s + 11·43-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 1/3·9-s + 0.301·11-s + 0.577·12-s − 1.66·13-s − 16-s − 1.69·17-s + 0.471·18-s + 1.14·19-s + 0.426·22-s + 0.208·23-s − 2.35·26-s + 0.192·27-s − 0.928·29-s + 1.43·31-s − 1.41·32-s + 0.174·33-s − 2.40·34-s + 1/3·36-s + 0.328·37-s + 1.62·38-s − 0.960·39-s − 1.87·41-s + 1.67·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.922813786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.922813786\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−14.70873791233132, −14.11400977971535, −13.78895900830563, −13.34530192751212, −12.83572927409317, −12.19516119657368, −11.93973844222173, −11.32499550657466, −10.76636646416507, −9.878701901944150, −9.581115832908204, −8.907779708586634, −8.465250812988647, −7.507130966622493, −7.128298234466632, −6.674218867687831, −5.912049780654457, −5.261412689973890, −4.789065777418282, −4.216631211309796, −3.758937868341699, −2.811568960469564, −2.562872142587585, −1.858162860212479, −0.5807647422109341,
0.5807647422109341, 1.858162860212479, 2.562872142587585, 2.811568960469564, 3.758937868341699, 4.216631211309796, 4.789065777418282, 5.261412689973890, 5.912049780654457, 6.674218867687831, 7.128298234466632, 7.507130966622493, 8.465250812988647, 8.907779708586634, 9.581115832908204, 9.878701901944150, 10.76636646416507, 11.32499550657466, 11.93973844222173, 12.19516119657368, 12.83572927409317, 13.34530192751212, 13.78895900830563, 14.11400977971535, 14.70873791233132
