| L(s) = 1 | + 9·3-s + 25·5-s + 16·7-s + 81·9-s + 564·11-s − 370·13-s + 225·15-s − 1.08e3·17-s + 2.86e3·19-s + 144·21-s − 1.58e3·23-s + 625·25-s + 729·27-s + 1.13e3·29-s + 6.01e3·31-s + 5.07e3·33-s + 400·35-s − 538·37-s − 3.33e3·39-s + 1.13e4·41-s − 5.44e3·43-s + 2.02e3·45-s − 1.02e4·47-s − 1.65e4·49-s − 9.77e3·51-s + 3.47e4·53-s + 1.41e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.123·7-s + 1/3·9-s + 1.40·11-s − 0.607·13-s + 0.258·15-s − 0.911·17-s + 1.81·19-s + 0.0712·21-s − 0.624·23-s + 1/5·25-s + 0.192·27-s + 0.250·29-s + 1.12·31-s + 0.811·33-s + 0.0551·35-s − 0.0646·37-s − 0.350·39-s + 1.05·41-s − 0.449·43-s + 0.149·45-s − 0.679·47-s − 0.984·49-s − 0.526·51-s + 1.69·53-s + 0.628·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.066887304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.066887304\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| 5 | \( 1 - p^{2} T \) |
| good | 7 | \( 1 - 16 T + p^{5} T^{2} \) |
| 11 | \( 1 - 564 T + p^{5} T^{2} \) |
| 13 | \( 1 + 370 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1086 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2860 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1584 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1134 T + p^{5} T^{2} \) |
| 31 | \( 1 - 6016 T + p^{5} T^{2} \) |
| 37 | \( 1 + 538 T + p^{5} T^{2} \) |
| 41 | \( 1 - 11370 T + p^{5} T^{2} \) |
| 43 | \( 1 + 5444 T + p^{5} T^{2} \) |
| 47 | \( 1 + 10296 T + p^{5} T^{2} \) |
| 53 | \( 1 - 34758 T + p^{5} T^{2} \) |
| 59 | \( 1 - 444 p T + p^{5} T^{2} \) |
| 61 | \( 1 - 9422 T + p^{5} T^{2} \) |
| 67 | \( 1 - 51124 T + p^{5} T^{2} \) |
| 71 | \( 1 + 14520 T + p^{5} T^{2} \) |
| 73 | \( 1 + 22678 T + p^{5} T^{2} \) |
| 79 | \( 1 - 97312 T + p^{5} T^{2} \) |
| 83 | \( 1 - 7956 T + p^{5} T^{2} \) |
| 89 | \( 1 + 47910 T + p^{5} T^{2} \) |
| 97 | \( 1 - 140738 T + p^{5} T^{2} \) |
| 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
−11.42301767736608461995942498287, −9.987558676509933452672230780102, −9.412696042519074687061490407866, −8.438532649381920531930545648601, −7.25968852256399647256154144359, −6.32193099252903968171523249394, −4.93764623011072924035068832408, −3.72984551184051202160127173545, −2.38924408112829935870829212525, −1.08997088270173144602412163569,
1.08997088270173144602412163569, 2.38924408112829935870829212525, 3.72984551184051202160127173545, 4.93764623011072924035068832408, 6.32193099252903968171523249394, 7.25968852256399647256154144359, 8.438532649381920531930545648601, 9.412696042519074687061490407866, 9.987558676509933452672230780102, 11.42301767736608461995942498287
