| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 4·7-s − 8-s + 9-s − 10-s − 2·11-s − 12-s − 13-s − 4·14-s − 15-s + 16-s − 2·17-s − 18-s − 2·19-s + 20-s − 4·21-s + 2·22-s − 7·23-s + 24-s + 25-s + 26-s − 27-s + 4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s − 0.288·12-s − 0.277·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.458·19-s + 0.223·20-s − 0.872·21-s + 0.426·22-s − 1.45·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−12.56984390542841, −12.16721472052645, −11.58524233230640, −11.30972835948800, −10.98002787823222, −10.35764615394850, −10.18583762708429, −9.557652222420429, −9.166834024913840, −8.373990713723034, −8.323361108174311, −7.615394683916541, −7.466712739619524, −6.683375012368706, −6.285438405461910, −5.726242393738592, −5.298314519713292, −4.883966219527110, −4.246341231872288, −3.899533663573042, −2.937874964605785, −2.191694196325502, −2.052698157040777, −1.420081990760234, −0.7059733791966814, 0,
0.7059733791966814, 1.420081990760234, 2.052698157040777, 2.191694196325502, 2.937874964605785, 3.899533663573042, 4.246341231872288, 4.883966219527110, 5.298314519713292, 5.726242393738592, 6.285438405461910, 6.683375012368706, 7.466712739619524, 7.615394683916541, 8.323361108174311, 8.373990713723034, 9.166834024913840, 9.557652222420429, 10.18583762708429, 10.35764615394850, 10.98002787823222, 11.30972835948800, 11.58524233230640, 12.16721472052645, 12.56984390542841
