L(s) = 1 | − 2-s − 4-s − 5-s + 7-s + 3·8-s + 10-s − 4·11-s − 4·13-s − 14-s − 16-s + 6·19-s + 20-s + 4·22-s + 6·23-s + 25-s + 4·26-s − 28-s + 4·31-s − 5·32-s − 35-s + 4·37-s − 6·38-s − 3·40-s − 2·41-s + 4·43-s + 4·44-s − 6·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 0.377·7-s + 1.06·8-s + 0.316·10-s − 1.20·11-s − 1.10·13-s − 0.267·14-s − 1/4·16-s + 1.37·19-s + 0.223·20-s + 0.852·22-s + 1.25·23-s + 1/5·25-s + 0.784·26-s − 0.188·28-s + 0.718·31-s − 0.883·32-s − 0.169·35-s + 0.657·37-s − 0.973·38-s − 0.474·40-s − 0.312·41-s + 0.609·43-s + 0.603·44-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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
−14.02820493592384, −13.64191595221333, −13.06708823158193, −12.70752067085507, −12.08756420463847, −11.59508219582999, −10.95630025331959, −10.66796334661793, −9.978970803323788, −9.642104277217349, −9.171348270640800, −8.572259612798964, −7.989987360546257, −7.584543231016178, −7.422107829982197, −6.647184996925933, −5.811261541263956, −5.067396775796301, −4.870862878566115, −4.458732001150475, −3.435856171427015, −2.988508518637211, −2.295115752954375, −1.406287032515943, −0.7394045502788123, 0,
0.7394045502788123, 1.406287032515943, 2.295115752954375, 2.988508518637211, 3.435856171427015, 4.458732001150475, 4.870862878566115, 5.067396775796301, 5.811261541263956, 6.647184996925933, 7.422107829982197, 7.584543231016178, 7.989987360546257, 8.572259612798964, 9.171348270640800, 9.642104277217349, 9.978970803323788, 10.66796334661793, 10.95630025331959, 11.59508219582999, 12.08756420463847, 12.70752067085507, 13.06708823158193, 13.64191595221333, 14.02820493592384