| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s − 14-s + 16-s + 20-s + 23-s + 25-s + 28-s − 29-s − 32-s + 35-s − 40-s − 41-s − 2·43-s − 46-s + 47-s − 50-s − 56-s + 58-s − 61-s + 64-s + 67-s − 70-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s − 14-s + 16-s + 20-s + 23-s + 25-s + 28-s − 29-s − 32-s + 35-s − 40-s − 41-s − 2·43-s − 46-s + 47-s − 50-s − 56-s + 58-s − 61-s + 64-s + 67-s − 70-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9776145696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9776145696\) |
| \(L(1)\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 - T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.531920694284459774476629745086, −8.829244695746261692429905605621, −8.180802378608011723867893344293, −7.26775723337616800611282048216, −6.55319609611195800513081400305, −5.59013981902223543531339556491, −4.87980830467915516407500973706, −3.31735302892291471283406407879, −2.16729579093822006075087991864, −1.37154530417691039299109025895,
1.37154530417691039299109025895, 2.16729579093822006075087991864, 3.31735302892291471283406407879, 4.87980830467915516407500973706, 5.59013981902223543531339556491, 6.55319609611195800513081400305, 7.26775723337616800611282048216, 8.180802378608011723867893344293, 8.829244695746261692429905605621, 9.531920694284459774476629745086
