L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 2·7-s + 8-s + 9-s + 2·11-s + 12-s − 13-s + 2·14-s + 16-s + 2·17-s + 18-s − 4·19-s + 2·21-s + 2·22-s + 24-s − 26-s + 27-s + 2·28-s + 4·29-s + 8·31-s + 32-s + 2·33-s + 2·34-s + 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.603·11-s + 0.288·12-s − 0.277·13-s + 0.534·14-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s + 0.436·21-s + 0.426·22-s + 0.204·24-s − 0.196·26-s + 0.192·27-s + 0.377·28-s + 0.742·29-s + 1.43·31-s + 0.176·32-s + 0.348·33-s + 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.871671394\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.871671394\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−9.092110735494385917277404308732, −8.261740578974282475628646606971, −7.71533725711756489273008899983, −6.70948077287911955851255894593, −6.05677467900280801452325604724, −4.85230961898891508041519458026, −4.37967211303981240246089391509, −3.33391362755583657763695615911, −2.38909622796055435807922194933, −1.32477871361042248821887123096,
1.32477871361042248821887123096, 2.38909622796055435807922194933, 3.33391362755583657763695615911, 4.37967211303981240246089391509, 4.85230961898891508041519458026, 6.05677467900280801452325604724, 6.70948077287911955851255894593, 7.71533725711756489273008899983, 8.261740578974282475628646606971, 9.092110735494385917277404308732