| L(s) = 1 | − 2-s + 4-s − 5-s − 2·7-s − 8-s + 10-s + 3·11-s − 13-s + 2·14-s + 16-s + 7·19-s − 20-s − 3·22-s − 23-s + 25-s + 26-s − 2·28-s − 29-s − 7·31-s − 32-s + 2·35-s − 11·37-s − 7·38-s + 40-s + 2·41-s − 43-s + 3·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s − 0.353·8-s + 0.316·10-s + 0.904·11-s − 0.277·13-s + 0.534·14-s + 1/4·16-s + 1.60·19-s − 0.223·20-s − 0.639·22-s − 0.208·23-s + 1/5·25-s + 0.196·26-s − 0.377·28-s − 0.185·29-s − 1.25·31-s − 0.176·32-s + 0.338·35-s − 1.80·37-s − 1.13·38-s + 0.158·40-s + 0.312·41-s − 0.152·43-s + 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 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
−12.65866233366132, −12.29432306613705, −11.83068321239214, −11.39712150337806, −11.10994308681212, −10.35537413802825, −10.08509880594917, −9.499985484251924, −9.204365539243107, −8.848872454856959, −8.235770830452907, −7.634526627372971, −7.385623677878392, −6.799248567344333, −6.507376040543086, −5.911150555606603, −5.165188741723800, −5.058081521225040, −3.987446847446634, −3.551381693374230, −3.382762472859023, −2.551237200187330, −1.957160985690600, −1.301263092822292, −0.6879983911901423, 0,
0.6879983911901423, 1.301263092822292, 1.957160985690600, 2.551237200187330, 3.382762472859023, 3.551381693374230, 3.987446847446634, 5.058081521225040, 5.165188741723800, 5.911150555606603, 6.507376040543086, 6.799248567344333, 7.385623677878392, 7.634526627372971, 8.235770830452907, 8.848872454856959, 9.204365539243107, 9.499985484251924, 10.08509880594917, 10.35537413802825, 11.10994308681212, 11.39712150337806, 11.83068321239214, 12.29432306613705, 12.65866233366132
