| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s − 2·9-s + 10-s − 2·11-s − 12-s − 5·13-s − 14-s − 15-s + 16-s − 5·17-s − 2·18-s + 5·19-s + 20-s + 21-s − 2·22-s + 6·23-s − 24-s + 25-s − 5·26-s + 5·27-s − 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 − 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.603·11-s − 0.288·12-s − 1.38·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 1.21·17-s − 0.471·18-s + 1.14·19-s + 0.223·20-s + 0.218·21-s − 0.426·22-s + 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.980·26-s + 0.962·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 139 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−15.01421276057299, −14.46890603123332, −13.94123150429132, −13.47231380753155, −12.96097406971704, −12.44658718232864, −12.01020560015985, −11.43463149464506, −10.87016143340534, −10.54967163983719, −9.826199407118914, −9.186500503026958, −8.886101821270891, −7.846301830077230, −7.387060732995906, −6.798621118898248, −6.327283208286218, −5.488125252280535, −5.314788638796500, −4.750334642878548, −4.028729744827346, −3.025402073102836, −2.710058863474472, −2.067571243985983, −0.9123344976497703, 0,
0.9123344976497703, 2.067571243985983, 2.710058863474472, 3.025402073102836, 4.028729744827346, 4.750334642878548, 5.314788638796500, 5.488125252280535, 6.327283208286218, 6.798621118898248, 7.387060732995906, 7.846301830077230, 8.886101821270891, 9.186500503026958, 9.826199407118914, 10.54967163983719, 10.87016143340534, 11.43463149464506, 12.01020560015985, 12.44658718232864, 12.96097406971704, 13.47231380753155, 13.94123150429132, 14.46890603123332, 15.01421276057299
