| L(s) = 1 | + 3-s − 7-s + 9-s + 3·11-s + 13-s + 3·17-s − 6·19-s − 21-s + 23-s + 27-s + 8·29-s + 4·31-s + 3·33-s − 5·37-s + 39-s − 5·41-s − 6·43-s + 8·47-s − 6·49-s + 3·51-s − 9·53-s − 6·57-s + 4·59-s − 11·61-s − 63-s − 16·67-s + 69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s + 0.727·17-s − 1.37·19-s − 0.218·21-s + 0.208·23-s + 0.192·27-s + 1.48·29-s + 0.718·31-s + 0.522·33-s − 0.821·37-s + 0.160·39-s − 0.780·41-s − 0.914·43-s + 1.16·47-s − 6/7·49-s + 0.420·51-s − 1.23·53-s − 0.794·57-s + 0.520·59-s − 1.40·61-s − 0.125·63-s − 1.95·67-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
| 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
−15.20452496459086, −14.89405287764761, −14.24155799651224, −13.82695287875504, −13.34366867747285, −12.71416423794825, −12.10084490984116, −11.89518696904166, −11.00094080118796, −10.31320814813421, −10.16248511399649, −9.178157183872183, −9.021894547686681, −8.197428703510875, −7.973438733268510, −6.922642088062120, −6.622267346731879, −6.099968391010571, −5.256023804456283, −4.452420524770176, −4.062075209330568, −3.170835335755333, −2.847680785313527, −1.758071000136861, −1.224469032840309, 0,
1.224469032840309, 1.758071000136861, 2.847680785313527, 3.170835335755333, 4.062075209330568, 4.452420524770176, 5.256023804456283, 6.099968391010571, 6.622267346731879, 6.922642088062120, 7.973438733268510, 8.197428703510875, 9.021894547686681, 9.178157183872183, 10.16248511399649, 10.31320814813421, 11.00094080118796, 11.89518696904166, 12.10084490984116, 12.71416423794825, 13.34366867747285, 13.82695287875504, 14.24155799651224, 14.89405287764761, 15.20452496459086
