| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s + 12-s − 13-s − 14-s + 15-s + 16-s + 17-s − 18-s + 5·19-s + 20-s + 21-s − 3·23-s − 24-s + 25-s + 26-s + 27-s + 28-s − 3·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.14·19-s + 0.223·20-s + 0.218·21-s − 0.625·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s − 0.557·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.182331839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.182331839\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.67638217717226, −13.19355684126266, −12.73199190837579, −12.05538496336393, −11.72447540997303, −11.13121189143117, −10.61105635790118, −10.13267910354687, −9.549663832590038, −9.326221184852871, −8.829194907221277, −8.138466929673498, −7.604179739423918, −7.526891896093217, −6.745536822738579, −6.076009259524598, −5.644418270185105, −5.080996940800354, −4.233477151332721, −3.879154485528115, −2.867943356012189, −2.635273443994216, −1.912127286089923, −1.225164838238428, −0.6404644260064002,
0.6404644260064002, 1.225164838238428, 1.912127286089923, 2.635273443994216, 2.867943356012189, 3.879154485528115, 4.233477151332721, 5.080996940800354, 5.644418270185105, 6.076009259524598, 6.745536822738579, 7.526891896093217, 7.604179739423918, 8.138466929673498, 8.829194907221277, 9.326221184852871, 9.549663832590038, 10.13267910354687, 10.61105635790118, 11.13121189143117, 11.72447540997303, 12.05538496336393, 12.73199190837579, 13.19355684126266, 13.67638217717226
