| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s + 2·11-s − 12-s − 6·13-s − 14-s + 16-s + 4·17-s + 18-s − 2·19-s + 21-s + 2·22-s + 2·23-s − 24-s − 6·26-s − 27-s − 28-s + 32-s − 2·33-s + 4·34-s + 36-s − 10·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.288·12-s − 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.458·19-s + 0.218·21-s + 0.426·22-s + 0.417·23-s − 0.204·24-s − 1.17·26-s − 0.192·27-s − 0.188·28-s + 0.176·32-s − 0.348·33-s + 0.685·34-s + 1/6·36-s − 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−13.97081317540994, −13.95303811400077, −12.97431449993910, −12.49267903829552, −12.26626335598915, −11.95140711450789, −11.25042340742477, −10.74065927689362, −10.21088324981950, −9.833121049993336, −9.223923757934249, −8.716994034673721, −7.811785672184927, −7.461988037192895, −6.846305018123765, −6.544415886505905, −5.801961770341068, −5.340424193406479, −4.858475076047142, −4.332619356257821, −3.610831935539245, −3.143675705485943, −2.364411477914549, −1.763614140130220, −0.8843127848869415, 0,
0.8843127848869415, 1.763614140130220, 2.364411477914549, 3.143675705485943, 3.610831935539245, 4.332619356257821, 4.858475076047142, 5.340424193406479, 5.801961770341068, 6.544415886505905, 6.846305018123765, 7.461988037192895, 7.811785672184927, 8.716994034673721, 9.223923757934249, 9.833121049993336, 10.21088324981950, 10.74065927689362, 11.25042340742477, 11.95140711450789, 12.26626335598915, 12.49267903829552, 12.97431449993910, 13.95303811400077, 13.97081317540994
