L(s) = 1 | + 2-s − 4-s − 2·5-s − 3·8-s − 2·10-s + 13-s − 16-s − 17-s + 8·19-s + 2·20-s + 6·23-s − 25-s + 26-s + 6·29-s + 8·31-s + 5·32-s − 34-s − 4·37-s + 8·38-s + 6·40-s − 6·41-s + 6·46-s − 8·47-s − 50-s − 52-s − 4·53-s + 6·58-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.06·8-s − 0.632·10-s + 0.277·13-s − 1/4·16-s − 0.242·17-s + 1.83·19-s + 0.447·20-s + 1.25·23-s − 1/5·25-s + 0.196·26-s + 1.11·29-s + 1.43·31-s + 0.883·32-s − 0.171·34-s − 0.657·37-s + 1.29·38-s + 0.948·40-s − 0.937·41-s + 0.884·46-s − 1.16·47-s − 0.141·50-s − 0.138·52-s − 0.549·53-s + 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.96833503769371, −13.62021496412891, −13.01742746666255, −12.72412661403916, −11.87609552492396, −11.70725318828346, −11.50672866061603, −10.54539484467696, −10.09780001809856, −9.573741358934874, −8.992171458218246, −8.473226982521058, −8.150156494209985, −7.387680234475282, −7.026733152999890, −6.279442213330029, −5.839769640050921, −4.990827623612199, −4.835108428863618, −4.273807111269786, −3.463496315276354, −3.205720409663190, −2.708104454026127, −1.472406044153157, −0.8453958889529547, 0,
0.8453958889529547, 1.472406044153157, 2.708104454026127, 3.205720409663190, 3.463496315276354, 4.273807111269786, 4.835108428863618, 4.990827623612199, 5.839769640050921, 6.279442213330029, 7.026733152999890, 7.387680234475282, 8.150156494209985, 8.473226982521058, 8.992171458218246, 9.573741358934874, 10.09780001809856, 10.54539484467696, 11.50672866061603, 11.70725318828346, 11.87609552492396, 12.72412661403916, 13.01742746666255, 13.62021496412891, 13.96833503769371