| L(s) = 1 | + 2-s + 4-s + 5-s + 3·7-s + 8-s + 10-s − 11-s + 13-s + 3·14-s + 16-s + 8·17-s − 5·19-s + 20-s − 22-s + 2·23-s + 25-s + 26-s + 3·28-s + 4·29-s + 5·31-s + 32-s + 8·34-s + 3·35-s − 5·38-s + 40-s − 2·41-s + 9·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.13·7-s + 0.353·8-s + 0.316·10-s − 0.301·11-s + 0.277·13-s + 0.801·14-s + 1/4·16-s + 1.94·17-s − 1.14·19-s + 0.223·20-s − 0.213·22-s + 0.417·23-s + 1/5·25-s + 0.196·26-s + 0.566·28-s + 0.742·29-s + 0.898·31-s + 0.176·32-s + 1.37·34-s + 0.507·35-s − 0.811·38-s + 0.158·40-s − 0.312·41-s + 1.37·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.130320381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.130320381\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−16.20516043021103, −15.63196921082544, −14.92986107825992, −14.52728542060559, −14.08757757792459, −13.60268451908233, −12.78441970775858, −12.42807340368913, −11.79173115687663, −11.15903882454808, −10.61625039292654, −10.11893686917672, −9.403663861581326, −8.476139627886458, −8.059244760107111, −7.493753779033322, −6.627544391264478, −6.011000529754036, −5.387735224642441, −4.814947413601861, −4.213978226292644, −3.285192536430962, −2.595974318623350, −1.705016795928306, −0.9856240148936440,
0.9856240148936440, 1.705016795928306, 2.595974318623350, 3.285192536430962, 4.213978226292644, 4.814947413601861, 5.387735224642441, 6.011000529754036, 6.627544391264478, 7.493753779033322, 8.059244760107111, 8.476139627886458, 9.403663861581326, 10.11893686917672, 10.61625039292654, 11.15903882454808, 11.79173115687663, 12.42807340368913, 12.78441970775858, 13.60268451908233, 14.08757757792459, 14.52728542060559, 14.92986107825992, 15.63196921082544, 16.20516043021103
