| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s − 2·9-s + 12-s + 13-s + 14-s + 16-s + 17-s + 2·18-s + 19-s − 21-s + 23-s − 24-s − 26-s − 5·27-s − 28-s − 3·29-s − 10·31-s − 32-s − 34-s − 2·36-s − 10·37-s − 38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.471·18-s + 0.229·19-s − 0.218·21-s + 0.208·23-s − 0.204·24-s − 0.196·26-s − 0.962·27-s − 0.188·28-s − 0.557·29-s − 1.79·31-s − 0.176·32-s − 0.171·34-s − 1/3·36-s − 1.64·37-s − 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.03596289429064, −13.25500555631889, −12.94715365280066, −12.39953237530706, −11.84423623671117, −11.32374842702288, −10.82127602142821, −10.53640498975312, −9.740303889929992, −9.364448074054887, −8.857369977416180, −8.703242137056154, −7.853851996188706, −7.502966465162152, −7.130357245299786, −6.329289117141906, −5.822724017444639, −5.471145452465535, −4.672501259745441, −3.818334503417230, −3.406019716706411, −2.914643203801778, −2.161732038993694, −1.702682642486221, −0.7627789873927054, 0,
0.7627789873927054, 1.702682642486221, 2.161732038993694, 2.914643203801778, 3.406019716706411, 3.818334503417230, 4.672501259745441, 5.471145452465535, 5.822724017444639, 6.329289117141906, 7.130357245299786, 7.502966465162152, 7.853851996188706, 8.703242137056154, 8.857369977416180, 9.364448074054887, 9.740303889929992, 10.53640498975312, 10.82127602142821, 11.32374842702288, 11.84423623671117, 12.39953237530706, 12.94715365280066, 13.25500555631889, 14.03596289429064
