| L(s) = 1 | − 5-s − 4·7-s − 6·11-s − 6·13-s + 3·17-s + 6·19-s − 7·23-s − 4·25-s − 6·31-s + 4·35-s − 3·37-s − 7·43-s − 6·47-s + 9·49-s − 10·53-s + 6·55-s − 6·59-s − 61-s + 6·65-s − 4·67-s − 7·71-s + 3·73-s + 24·77-s + 2·79-s + 15·83-s − 3·85-s − 9·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s − 1.80·11-s − 1.66·13-s + 0.727·17-s + 1.37·19-s − 1.45·23-s − 4/5·25-s − 1.07·31-s + 0.676·35-s − 0.493·37-s − 1.06·43-s − 0.875·47-s + 9/7·49-s − 1.37·53-s + 0.809·55-s − 0.781·59-s − 0.128·61-s + 0.744·65-s − 0.488·67-s − 0.830·71-s + 0.351·73-s + 2.73·77-s + 0.225·79-s + 1.64·83-s − 0.325·85-s − 0.953·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.21793996264369532199224244864, −6.44565603128433895413307230860, −5.45875951406284077765929146067, −5.26175676518936530355797830268, −4.17337230719408802639730485316, −3.20648596077886558956405360617, −2.92238247327108965304336587330, −1.88712662103680512819280047071, 0, 0,
1.88712662103680512819280047071, 2.92238247327108965304336587330, 3.20648596077886558956405360617, 4.17337230719408802639730485316, 5.26175676518936530355797830268, 5.45875951406284077765929146067, 6.44565603128433895413307230860, 7.21793996264369532199224244864
