| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s − 11-s − 12-s − 4·13-s − 14-s + 16-s + 6·17-s + 18-s + 21-s − 22-s − 2·23-s − 24-s − 4·26-s − 27-s − 28-s + 8·29-s − 10·31-s + 32-s + 33-s + 6·34-s + 36-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 − 0.301·11-s − 0.288·12-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.218·21-s − 0.213·22-s − 0.417·23-s − 0.204·24-s − 0.784·26-s − 0.192·27-s − 0.188·28-s + 1.48·29-s − 1.79·31-s + 0.176·32-s + 0.174·33-s + 1.02·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−16.43468391664822, −16.31800531934186, −15.60919368816158, −14.88040995836480, −14.49439503842296, −13.89311586690473, −13.24237326944717, −12.53762487866876, −12.20597796360272, −11.84112583171644, −10.94828049418734, −10.35889062245489, −9.978197627675973, −9.257382883204158, −8.382553246746819, −7.494273625611188, −7.241440513484038, −6.422814044569666, −5.656422373907697, −5.315654393504094, −4.568467760732934, −3.787260475922487, −3.050397677111413, −2.272226431706206, −1.221604289369816, 0,
1.221604289369816, 2.272226431706206, 3.050397677111413, 3.787260475922487, 4.568467760732934, 5.315654393504094, 5.656422373907697, 6.422814044569666, 7.241440513484038, 7.494273625611188, 8.382553246746819, 9.257382883204158, 9.978197627675973, 10.35889062245489, 10.94828049418734, 11.84112583171644, 12.20597796360272, 12.53762487866876, 13.24237326944717, 13.89311586690473, 14.49439503842296, 14.88040995836480, 15.60919368816158, 16.31800531934186, 16.43468391664822
