| L(s) = 1 | + 2·2-s + 3-s + 3·4-s − 5-s + 2·6-s − 7-s + 4·8-s + 9-s − 2·10-s + 3·12-s − 13-s − 2·14-s − 15-s + 5·16-s − 17-s + 2·18-s + 2·19-s − 3·20-s − 21-s − 23-s + 4·24-s − 2·26-s + 27-s − 3·28-s − 2·30-s − 31-s + 6·32-s + ⋯ |
| L(s) = 1 | + 2·2-s + 3-s + 3·4-s − 5-s + 2·6-s − 7-s + 4·8-s + 9-s − 2·10-s + 3·12-s − 13-s − 2·14-s − 15-s + 5·16-s − 17-s + 2·18-s + 2·19-s − 3·20-s − 21-s − 23-s + 4·24-s − 2·26-s + 27-s − 3·28-s − 2·30-s − 31-s + 6·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.987693778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.987693778\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 1213 | \( 1 - T \) |
| good | 2 | \( ( 1 - T )^{2} \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( ( 1 - T )^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
| 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
−8.372309953383061799009924823699, −7.59694210948673463339070688723, −7.11662048640602400248761190395, −6.57981772631958159129305105039, −5.46186510376505388231796284411, −4.74232042248841882239469793333, −3.88600230177775219405026227930, −3.44056410411462722857714575385, −2.78158939791588371638951911035, −1.83762089463751450073571834195,
1.83762089463751450073571834195, 2.78158939791588371638951911035, 3.44056410411462722857714575385, 3.88600230177775219405026227930, 4.74232042248841882239469793333, 5.46186510376505388231796284411, 6.57981772631958159129305105039, 7.11662048640602400248761190395, 7.59694210948673463339070688723, 8.372309953383061799009924823699
