| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s − 11-s + 12-s + 13-s + 14-s + 16-s + 17-s − 18-s + 19-s − 21-s + 22-s + 23-s − 24-s + 25-s − 26-s + 27-s − 28-s − 32-s − 33-s − 34-s + 36-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s − 11-s + 12-s + 13-s + 14-s + 16-s + 17-s − 18-s + 19-s − 21-s + 22-s + 23-s − 24-s + 25-s − 26-s + 27-s − 28-s − 32-s − 33-s − 34-s + 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8732271649\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8732271649\) |
| \(L(1)\) |
|
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 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 + T )^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
| 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
−10.17070288411288966292365777632, −9.404552424873271674373047695723, −8.733993683782842667696985210461, −7.939679133256527488033794483162, −7.19543378968230026663070994594, −6.37083535822805572840255488502, −5.16199790886719999902948233251, −3.25713902259192698288778659916, −3.07344589678601284797082503834, −1.41506081603072603018522033344,
1.41506081603072603018522033344, 3.07344589678601284797082503834, 3.25713902259192698288778659916, 5.16199790886719999902948233251, 6.37083535822805572840255488502, 7.19543378968230026663070994594, 7.939679133256527488033794483162, 8.733993683782842667696985210461, 9.404552424873271674373047695723, 10.17070288411288966292365777632
