| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 4·7-s + 8-s − 2·9-s + 10-s − 11-s + 12-s − 13-s + 4·14-s + 15-s + 16-s + 17-s − 2·18-s + 3·19-s + 20-s + 4·21-s − 22-s + 4·23-s + 24-s + 25-s − 26-s − 5·27-s + 4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.51·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.301·11-s + 0.288·12-s − 0.277·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.471·18-s + 0.688·19-s + 0.223·20-s + 0.872·21-s − 0.213·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.962·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.062691327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.062691327\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−9.103012589818820410649420956528, −8.320789511186694692935191320463, −7.75207319723562503709582073052, −6.90303904163221676218740503085, −5.71909843611383028416257810756, −5.18862703822122146231150076515, −4.41831679463306347328383675326, −3.19858249719584221378025866955, −2.43238690622000985858603227220, −1.39303663972292066318624139099,
1.39303663972292066318624139099, 2.43238690622000985858603227220, 3.19858249719584221378025866955, 4.41831679463306347328383675326, 5.18862703822122146231150076515, 5.71909843611383028416257810756, 6.90303904163221676218740503085, 7.75207319723562503709582073052, 8.320789511186694692935191320463, 9.103012589818820410649420956528
