| L(s) = 1 | + 32·2-s + 758.·3-s + 1.02e3·4-s − 3.06e3·5-s + 2.42e4·6-s + 2.76e4·7-s + 3.27e4·8-s + 3.98e5·9-s − 9.79e4·10-s + 1.61e5·11-s + 7.76e5·12-s − 1.14e6·13-s + 8.84e5·14-s − 2.32e6·15-s + 1.04e6·16-s − 5.38e6·17-s + 1.27e7·18-s + 1.40e7·19-s − 3.13e6·20-s + 2.09e7·21-s + 5.15e6·22-s − 8.08e6·23-s + 2.48e7·24-s − 3.94e7·25-s − 3.64e7·26-s + 1.67e8·27-s + 2.82e7·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.80·3-s + 0.5·4-s − 0.438·5-s + 1.27·6-s + 0.621·7-s + 0.353·8-s + 2.24·9-s − 0.309·10-s + 0.301·11-s + 0.901·12-s − 0.851·13-s + 0.439·14-s − 0.789·15-s + 0.250·16-s − 0.920·17-s + 1.58·18-s + 1.30·19-s − 0.219·20-s + 1.11·21-s + 0.213·22-s − 0.261·23-s + 0.637·24-s − 0.808·25-s − 0.602·26-s + 2.24·27-s + 0.310·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(4.990417653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.990417653\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 32T \) |
| 11 | \( 1 - 1.61e5T \) |
| good | 3 | \( 1 - 758.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 3.06e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 2.76e4T + 1.97e9T^{2} \) |
| 13 | \( 1 + 1.14e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 5.38e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.40e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 8.08e6T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.50e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.28e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 3.35e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 2.32e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.21e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.50e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.04e8T + 9.26e18T^{2} \) |
| 59 | \( 1 - 1.04e10T + 3.01e19T^{2} \) |
| 61 | \( 1 + 5.90e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 7.12e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.64e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 3.48e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 4.29e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 9.44e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + 3.81e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 7.29e10T + 7.15e21T^{2} \) |
| 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
−14.95967038171954412297043027438, −14.23492718877414095568248028503, −13.18658306852264229913976700736, −11.69816115483870621446881881832, −9.746444616624395454369883942403, −8.285534677550182829511386381205, −7.21245946944998582945189400112, −4.59076764109185162639807160345, −3.25698629922027330967379594373, −1.87376813686508858118463948080,
1.87376813686508858118463948080, 3.25698629922027330967379594373, 4.59076764109185162639807160345, 7.21245946944998582945189400112, 8.285534677550182829511386381205, 9.746444616624395454369883942403, 11.69816115483870621446881881832, 13.18658306852264229913976700736, 14.23492718877414095568248028503, 14.95967038171954412297043027438
