| L(s) = 1 | + 2-s − 3·3-s + 4-s − 5-s − 3·6-s − 5·7-s + 8-s + 6·9-s − 10-s − 4·11-s − 3·12-s − 13-s − 5·14-s + 3·15-s + 16-s − 3·17-s + 6·18-s + 19-s − 20-s + 15·21-s − 4·22-s + 7·23-s − 3·24-s + 25-s − 26-s − 9·27-s − 5·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.447·5-s − 1.22·6-s − 1.88·7-s + 0.353·8-s + 2·9-s − 0.316·10-s − 1.20·11-s − 0.866·12-s − 0.277·13-s − 1.33·14-s + 0.774·15-s + 1/4·16-s − 0.727·17-s + 1.41·18-s + 0.229·19-s − 0.223·20-s + 3.27·21-s − 0.852·22-s + 1.45·23-s − 0.612·24-s + 1/5·25-s − 0.196·26-s − 1.73·27-s − 0.944·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−12.32630247841887093527540219554, −11.11517582423130592220152114812, −10.53685995532109798031487935325, −9.431192927280394698141787607684, −7.34314124395432920163711504829, −6.59012976667376581440253973920, −5.66768121415484817418669355380, −4.67101925823320154679184379858, −3.15015586052048758770880964700, 0,
3.15015586052048758770880964700, 4.67101925823320154679184379858, 5.66768121415484817418669355380, 6.59012976667376581440253973920, 7.34314124395432920163711504829, 9.431192927280394698141787607684, 10.53685995532109798031487935325, 11.11517582423130592220152114812, 12.32630247841887093527540219554
