L(s) = 1 | − 6.00i·2-s − 4.04·4-s − 54.7·5-s + 232. i·7-s − 167. i·8-s + 328. i·10-s − 132.·11-s + 1.03e3·13-s + 1.39e3·14-s − 1.13e3·16-s − 771.·17-s − 2.41e3i·19-s + 221.·20-s + 794. i·22-s + (−14.4 − 2.53e3i)23-s + ⋯ |
L(s) = 1 | − 1.06i·2-s − 0.126·4-s − 0.979·5-s + 1.79i·7-s − 0.927i·8-s + 1.04i·10-s − 0.329·11-s + 1.69·13-s + 1.90·14-s − 1.11·16-s − 0.647·17-s − 1.53i·19-s + 0.123·20-s + 0.349i·22-s + (−0.00569 − 0.999i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.339977153\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.339977153\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 + (14.4 + 2.53e3i)T \) |
good | 2 | \( 1 + 6.00iT - 32T^{2} \) |
| 5 | \( 1 + 54.7T + 3.12e3T^{2} \) |
| 7 | \( 1 - 232. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 132.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.03e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 771.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.41e3iT - 2.47e6T^{2} \) |
| 29 | \( 1 - 2.49e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 4.50e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.87e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.88e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 3.10e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 5.24e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 3.02e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.19e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 2.65e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 5.42e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 93.9iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 3.90e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.13e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 7.24e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.22e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.63e5iT - 8.58e9T^{2} \) |
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
−11.29835658533848069214865111309, −10.55205276893642818970147640563, −8.950248873947803409069604597357, −8.616966249420767045250339331174, −6.99414238511321934612193712078, −5.83279523439113797045558224978, −4.31428294960634316315943898027, −3.08415162599445325720225914308, −2.13642215866107986324781289173, −0.44116562964048617780494659616,
1.21804005325066940245042648172, 3.55305602634603758362636719868, 4.40356488795152408498906024846, 5.99579543096903135407563330966, 6.90375966977245281900406045100, 7.909819278622732497695407080897, 8.241390061459321714962451257614, 10.03543145808772072877141935992, 11.05418393715846842798565468413, 11.62230621409748299726173364178