| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 4·7-s + 8-s + 9-s − 10-s − 4·11-s − 12-s − 2·13-s − 4·14-s + 15-s + 16-s + 17-s + 18-s − 4·19-s − 20-s + 4·21-s − 4·22-s − 4·23-s − 24-s + 25-s − 2·26-s − 27-s − 4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s − 0.288·12-s − 0.554·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.872·21-s − 0.852·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−10.22799198314520600987512889691, −10.15893096542881929011711805517, −8.582150486484030063688792843390, −7.44642620066667323155476644258, −6.61505027276764663344217644210, −5.78116542740457073968544189668, −4.75829110517886110282824961706, −3.62747450824378703746901074028, −2.53757393131658043958694940113, 0,
2.53757393131658043958694940113, 3.62747450824378703746901074028, 4.75829110517886110282824961706, 5.78116542740457073968544189668, 6.61505027276764663344217644210, 7.44642620066667323155476644258, 8.582150486484030063688792843390, 10.15893096542881929011711805517, 10.22799198314520600987512889691
