| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 5·7-s + 8-s − 2·9-s + 10-s + 11-s + 12-s + 4·13-s + 5·14-s + 15-s + 16-s + 17-s − 2·18-s + 7·19-s + 20-s + 5·21-s + 22-s + 8·23-s + 24-s + 25-s + 4·26-s − 5·27-s + 5·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.88·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s + 1.10·13-s + 1.33·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.471·18-s + 1.60·19-s + 0.223·20-s + 1.09·21-s + 0.213·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.784·26-s − 0.962·27-s + 0.944·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.590070692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.590070692\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 11 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.293635854471329195384609932348, −7.53375844824489400308132087217, −6.99024279853805331316094490451, −5.70039217147278135701040872176, −5.41370680067114110948002765045, −4.72246126695946412008487609416, −3.58198890718384045082579882256, −3.11570918719347193540994950175, −1.81571818368577660326243258448, −1.39559021097440837064201338799,
1.39559021097440837064201338799, 1.81571818368577660326243258448, 3.11570918719347193540994950175, 3.58198890718384045082579882256, 4.72246126695946412008487609416, 5.41370680067114110948002765045, 5.70039217147278135701040872176, 6.99024279853805331316094490451, 7.53375844824489400308132087217, 8.293635854471329195384609932348
