L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s − 5·11-s + 12-s + 13-s − 14-s + 16-s − 5·17-s − 18-s + 21-s + 5·22-s − 24-s − 26-s + 27-s + 28-s − 7·29-s − 9·31-s − 32-s − 5·33-s + 5·34-s + 36-s + 8·37-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 + 1/3·9-s − 1.50·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 1.21·17-s − 0.235·18-s + 0.218·21-s + 1.06·22-s − 0.204·24-s − 0.196·26-s + 0.192·27-s + 0.188·28-s − 1.29·29-s − 1.61·31-s − 0.176·32-s − 0.870·33-s + 0.857·34-s + 1/6·36-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 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
−8.838873211957443440524404075738, −7.950827946110507622049945061588, −7.59479972503915351853035329473, −6.63954289483917080475778683887, −5.61896013855519810560445379067, −4.73644888182386750015652030805, −3.59490695585792095415121156028, −2.53805162742335217943346409509, −1.75231505745576186522341502521, 0,
1.75231505745576186522341502521, 2.53805162742335217943346409509, 3.59490695585792095415121156028, 4.73644888182386750015652030805, 5.61896013855519810560445379067, 6.63954289483917080475778683887, 7.59479972503915351853035329473, 7.950827946110507622049945061588, 8.838873211957443440524404075738