| L(s) = 1 | − 2-s − 3-s + 5-s + 6-s − 7-s + 8-s + 9-s − 10-s + 11-s + 13-s + 14-s − 15-s − 16-s − 18-s + 19-s + 21-s − 22-s − 24-s − 26-s − 27-s − 29-s + 30-s − 33-s − 35-s − 37-s − 38-s − 39-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 5-s + 6-s − 7-s + 8-s + 9-s − 10-s + 11-s + 13-s + 14-s − 15-s − 16-s − 18-s + 19-s + 21-s − 22-s − 24-s − 26-s − 27-s − 29-s + 30-s − 33-s − 35-s − 37-s − 38-s − 39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3701288784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3701288784\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 5 | \( 1 - T + T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( ( 1 + T )^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 + T )^{2} \) |
| 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
−12.32848537555585428969738239349, −11.19890634879836368059449488540, −10.28003921063227548682425678416, −9.561448995802831927557230678629, −8.955682220067399655377402410399, −7.33965796238802130019399168707, −6.37089988384106126433373235250, −5.49452683200611078436646030950, −3.89212545776047806075019712736, −1.42432380248619842007983818178,
1.42432380248619842007983818178, 3.89212545776047806075019712736, 5.49452683200611078436646030950, 6.37089988384106126433373235250, 7.33965796238802130019399168707, 8.955682220067399655377402410399, 9.561448995802831927557230678629, 10.28003921063227548682425678416, 11.19890634879836368059449488540, 12.32848537555585428969738239349
