L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 3·7-s − 8-s − 2·9-s − 2·11-s + 12-s + 13-s + 3·14-s + 16-s + 2·18-s − 2·19-s − 3·21-s + 2·22-s − 3·23-s − 24-s − 26-s − 5·27-s − 3·28-s − 2·29-s − 32-s − 2·33-s − 2·36-s + 2·37-s + 2·38-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.13·7-s − 0.353·8-s − 2/3·9-s − 0.603·11-s + 0.288·12-s + 0.277·13-s + 0.801·14-s + 1/4·16-s + 0.471·18-s − 0.458·19-s − 0.654·21-s + 0.426·22-s − 0.625·23-s − 0.204·24-s − 0.196·26-s − 0.962·27-s − 0.566·28-s − 0.371·29-s − 0.176·32-s − 0.348·33-s − 1/3·36-s + 0.328·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−13.57974108226108, −13.08400790829576, −12.75438264200890, −12.16046061366585, −11.62953836522245, −11.18831417594014, −10.69059297722033, −10.09473033071173, −9.821969459279263, −9.299141694462717, −8.890746627706582, −8.253982579000949, −8.095442368733392, −7.510466160860488, −6.779452035785237, −6.494594473930420, −5.934137266982417, −5.432682847651170, −4.809424646843224, −3.932217608273651, −3.453687585149725, −3.057606117706785, −2.394646315975501, −1.997033456487024, −1.115306608248534, 0, 0,
1.115306608248534, 1.997033456487024, 2.394646315975501, 3.057606117706785, 3.453687585149725, 3.932217608273651, 4.809424646843224, 5.432682847651170, 5.934137266982417, 6.494594473930420, 6.779452035785237, 7.510466160860488, 8.095442368733392, 8.253982579000949, 8.890746627706582, 9.299141694462717, 9.821969459279263, 10.09473033071173, 10.69059297722033, 11.18831417594014, 11.62953836522245, 12.16046061366585, 12.75438264200890, 13.08400790829576, 13.57974108226108