| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 3·7-s − 8-s − 2·9-s − 10-s − 6·11-s − 12-s + 3·13-s + 3·14-s − 15-s + 16-s + 17-s + 2·18-s − 7·19-s + 20-s + 3·21-s + 6·22-s + 4·23-s + 24-s + 25-s − 3·26-s + 5·27-s − 3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s − 1.80·11-s − 0.288·12-s + 0.832·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.471·18-s − 1.60·19-s + 0.223·20-s + 0.654·21-s + 1.27·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.588·26-s + 0.962·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 139 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−15.39986482468828, −15.05939755694288, −14.18038496008135, −13.67627008865563, −13.01190323909327, −12.66997733889101, −12.38447636853684, −11.37143797984180, −10.82515833397532, −10.69638442946828, −10.18290990753825, −9.462074792760568, −8.935044983808752, −8.509498344518027, −7.867372659715754, −7.218424824574339, −6.596878715119223, −6.024941491804726, −5.716792304454140, −5.100811958050608, −4.255173244254103, −3.261968387186028, −2.841652047359148, −2.218052167572650, −1.238328488165474, 0, 0,
1.238328488165474, 2.218052167572650, 2.841652047359148, 3.261968387186028, 4.255173244254103, 5.100811958050608, 5.716792304454140, 6.024941491804726, 6.596878715119223, 7.218424824574339, 7.867372659715754, 8.509498344518027, 8.935044983808752, 9.462074792760568, 10.18290990753825, 10.69638442946828, 10.82515833397532, 11.37143797984180, 12.38447636853684, 12.66997733889101, 13.01190323909327, 13.67627008865563, 14.18038496008135, 15.05939755694288, 15.39986482468828
