| 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 − 3·13-s − 14-s − 15-s + 16-s + 3·17-s + 2·18-s + 5·19-s + 20-s − 21-s + 2·22-s − 8·23-s + 24-s + 25-s + 3·26-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 − 0.832·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.471·18-s + 1.14·19-s + 0.223·20-s − 0.218·21-s + 0.426·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.588·26-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 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−15.02441435350372, −14.36454637340213, −14.26869387552900, −13.48069552078697, −12.89350086028706, −12.12930338906603, −11.86327899164197, −11.44750173874331, −10.75731023827722, −10.18894615395483, −9.877191541540666, −9.375468216557787, −8.543729257315384, −8.106040516958536, −7.672208609513371, −6.948153384777606, −6.436612250168680, −5.623832166109928, −5.339886826154973, −4.886883858759759, −3.755366972030257, −3.124489007492020, −2.333415420079221, −1.790778367125152, −0.8088758678057277, 0,
0.8088758678057277, 1.790778367125152, 2.333415420079221, 3.124489007492020, 3.755366972030257, 4.886883858759759, 5.339886826154973, 5.623832166109928, 6.436612250168680, 6.948153384777606, 7.672208609513371, 8.106040516958536, 8.543729257315384, 9.375468216557787, 9.877191541540666, 10.18894615395483, 10.75731023827722, 11.44750173874331, 11.86327899164197, 12.12930338906603, 12.89350086028706, 13.48069552078697, 14.26869387552900, 14.36454637340213, 15.02441435350372
