L(s) = 1 | + 2-s + 4-s + 5-s + 8-s − 3·9-s + 10-s + 4·11-s + 6·13-s + 16-s − 2·17-s − 3·18-s + 4·19-s + 20-s + 4·22-s + 2·23-s + 25-s + 6·26-s − 10·29-s − 8·31-s + 32-s − 2·34-s − 3·36-s + 10·37-s + 4·38-s + 40-s − 6·41-s − 6·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s − 9-s + 0.316·10-s + 1.20·11-s + 1.66·13-s + 1/4·16-s − 0.485·17-s − 0.707·18-s + 0.917·19-s + 0.223·20-s + 0.852·22-s + 0.417·23-s + 1/5·25-s + 1.17·26-s − 1.85·29-s − 1.43·31-s + 0.176·32-s − 0.342·34-s − 1/2·36-s + 1.64·37-s + 0.648·38-s + 0.158·40-s − 0.937·41-s − 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 137 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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
−13.90045711243297, −13.41896181590764, −13.22876911371604, −12.73817180731958, −11.88446301643248, −11.44323090182202, −11.29355075991724, −10.86309698527572, −10.06199595036346, −9.430920470282645, −9.076269111551880, −8.601523354550034, −8.042340484738109, −7.317782426454501, −6.801108541114982, −6.213877090811198, −5.884780688042316, −5.421945562567906, −4.795337927401201, −4.037717783328260, −3.435669292641819, −3.295616682213946, −2.321023292765027, −1.604370107970545, −1.191754201292775, 0,
1.191754201292775, 1.604370107970545, 2.321023292765027, 3.295616682213946, 3.435669292641819, 4.037717783328260, 4.795337927401201, 5.421945562567906, 5.884780688042316, 6.213877090811198, 6.801108541114982, 7.317782426454501, 8.042340484738109, 8.601523354550034, 9.076269111551880, 9.430920470282645, 10.06199595036346, 10.86309698527572, 11.29355075991724, 11.44323090182202, 11.88446301643248, 12.73817180731958, 13.22876911371604, 13.41896181590764, 13.90045711243297