| L(s) = 1 | + 2-s − 4-s − 3·8-s − 3·9-s − 3·11-s + 13-s − 16-s − 7·17-s − 3·18-s + 7·19-s − 3·22-s − 6·23-s − 5·25-s + 26-s − 5·29-s + 5·32-s − 7·34-s + 3·36-s + 8·37-s + 7·38-s + 2·43-s + 3·44-s − 6·46-s − 7·47-s − 5·50-s − 52-s − 3·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 9-s − 0.904·11-s + 0.277·13-s − 1/4·16-s − 1.69·17-s − 0.707·18-s + 1.60·19-s − 0.639·22-s − 1.25·23-s − 25-s + 0.196·26-s − 0.928·29-s + 0.883·32-s − 1.20·34-s + 1/2·36-s + 1.31·37-s + 1.13·38-s + 0.304·43-s + 0.452·44-s − 0.884·46-s − 1.02·47-s − 0.707·50-s − 0.138·52-s − 0.412·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−10.07084765051358713543920425978, −9.259730413329108464669848282618, −8.427602051723064213968857703075, −7.58221538537525761723731254609, −6.13619288958387853257590270753, −5.56220221892590651265131476874, −4.57403153778540150751986491981, −3.52729931218747647603184716659, −2.43258577292117985821674468072, 0,
2.43258577292117985821674468072, 3.52729931218747647603184716659, 4.57403153778540150751986491981, 5.56220221892590651265131476874, 6.13619288958387853257590270753, 7.58221538537525761723731254609, 8.427602051723064213968857703075, 9.259730413329108464669848282618, 10.07084765051358713543920425978
