| L(s) = 1 | − 11.9i·2-s + 44.5·3-s − 14.1·4-s + 221.·5-s − 531. i·6-s + 241. i·7-s − 1.35e3i·8-s − 200.·9-s − 2.64e3i·10-s + 714. i·11-s − 629.·12-s + 1.12e4·13-s + 2.88e3·14-s + 9.88e3·15-s − 1.79e4·16-s − 3.04e4i·17-s + ⋯ |
| L(s) = 1 | − 1.05i·2-s + 0.952·3-s − 0.110·4-s + 0.793·5-s − 1.00i·6-s + 0.266i·7-s − 0.937i·8-s − 0.0918·9-s − 0.836i·10-s + 0.161i·11-s − 0.105·12-s + 1.42·13-s + 0.280·14-s + 0.756·15-s − 1.09·16-s − 1.50i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0821 + 0.996i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.0821 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.24261 - 2.43508i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.24261 - 2.43508i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 61 | \( 1 + (1.45e5 - 1.76e6i)T \) |
| good | 2 | \( 1 + 11.9iT - 128T^{2} \) |
| 3 | \( 1 - 44.5T + 2.18e3T^{2} \) |
| 5 | \( 1 - 221.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 241. iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 714. iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 1.12e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.04e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 5.30e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.03e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 1.43e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 1.58e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 + 4.56e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 7.77e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.58e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 1.99e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.87e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 1.26e6iT - 2.48e12T^{2} \) |
| 67 | \( 1 + 3.80e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 1.73e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 + 5.63e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.19e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 - 4.92e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.65e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 + 9.89e6T + 8.07e13T^{2} \) |
| 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
−13.56766889696599032285356082943, −12.00362676880514140064547293766, −11.07026633236505306872805402173, −9.664708387984360622848759436266, −9.059298883958435810695642358628, −7.38083431105500720492351023817, −5.67998165535063191483216988493, −3.51872431893369984869537920746, −2.54509414522260836725105137788, −1.24912234634721019993957486259,
1.75856166601985239610236620538, 3.41191937314907513153698002589, 5.54672254835896231597574585082, 6.53391163915707544443386272210, 8.037984280719206695426472854268, 8.716017800620250433005993805354, 10.16174994816240102463393479657, 11.56561266977009817140358474667, 13.53179583643508404225671461829, 13.86272477979053536460690087696
