| L(s) = 1 | − 2-s − 3-s − 4-s − 3·5-s + 6-s − 3·7-s + 3·8-s + 9-s + 3·10-s − 2·11-s + 12-s − 4·13-s + 3·14-s + 3·15-s − 16-s − 7·17-s − 18-s − 5·19-s + 3·20-s + 3·21-s + 2·22-s − 6·23-s − 3·24-s + 4·25-s + 4·26-s − 27-s + 3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 1.34·5-s + 0.408·6-s − 1.13·7-s + 1.06·8-s + 1/3·9-s + 0.948·10-s − 0.603·11-s + 0.288·12-s − 1.10·13-s + 0.801·14-s + 0.774·15-s − 1/4·16-s − 1.69·17-s − 0.235·18-s − 1.14·19-s + 0.670·20-s + 0.654·21-s + 0.426·22-s − 1.25·23-s − 0.612·24-s + 4/5·25-s + 0.784·26-s − 0.192·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1077 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1077 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 359 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.169870359820796821768170226854, −8.194200788035384413463948972920, −7.49544883671035519427210080537, −6.78570590217135349517649462153, −5.63477906791979445906843426847, −4.35197758069055135672381002336, −4.02464636813567350786306115869, −2.37115444275316423993424685640, 0, 0,
2.37115444275316423993424685640, 4.02464636813567350786306115869, 4.35197758069055135672381002336, 5.63477906791979445906843426847, 6.78570590217135349517649462153, 7.49544883671035519427210080537, 8.194200788035384413463948972920, 9.169870359820796821768170226854
