| L(s) = 1 | + (1.27 + 0.738i)5-s + (4.08 + 5.68i)7-s + (7.76 + 13.4i)11-s + 18.6i·13-s + (−11.8 + 6.81i)17-s + (−1.20 − 0.696i)19-s + (20.0 − 34.6i)23-s + (−11.4 − 19.7i)25-s − 52.1·29-s + (−41.5 + 24.0i)31-s + (1.02 + 10.2i)35-s + (−7.93 + 13.7i)37-s − 20.6i·41-s + 16.0·43-s + (65.1 + 37.6i)47-s + ⋯ |
| L(s) = 1 | + (0.255 + 0.147i)5-s + (0.583 + 0.811i)7-s + (0.705 + 1.22i)11-s + 1.43i·13-s + (−0.694 + 0.400i)17-s + (−0.0635 − 0.0366i)19-s + (0.870 − 1.50i)23-s + (−0.456 − 0.790i)25-s − 1.79·29-s + (−1.34 + 0.774i)31-s + (0.0293 + 0.293i)35-s + (−0.214 + 0.371i)37-s − 0.503i·41-s + 0.373·43-s + (1.38 + 0.800i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.476 - 0.879i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.476 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.699404913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.699404913\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-4.08 - 5.68i)T \) |
| good | 5 | \( 1 + (-1.27 - 0.738i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (-7.76 - 13.4i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 18.6iT - 169T^{2} \) |
| 17 | \( 1 + (11.8 - 6.81i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (1.20 + 0.696i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-20.0 + 34.6i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 52.1T + 841T^{2} \) |
| 31 | \( 1 + (41.5 - 24.0i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (7.93 - 13.7i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 20.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 16.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-65.1 - 37.6i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (9.13 + 15.8i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-41.7 + 24.0i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-23.2 - 13.4i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-0.310 - 0.537i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 18.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + (89.5 - 51.7i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (55.1 - 95.4i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 23.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (114. + 66.1i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 72.0iT - 9.40e3T^{2} \) |
| 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
−9.940789506949362571716926051925, −8.988667336828677722001146133016, −8.771776975930554897066307640259, −7.33782138530969068729048781948, −6.74628996065771828786448983493, −5.80220021892861306486499989354, −4.68312567520384744350988666426, −4.03878974108925573254452631278, −2.32749651334357679306110382853, −1.73603694574576617747006657342,
0.51957240817300006388533846855, 1.67876874860731797360117610716, 3.27485467514658239616961981321, 3.99147693719299931794193295306, 5.40245923452218139251388167235, 5.77808178711734029864828509879, 7.24294938120257609675159387061, 7.64308488293836421355266837018, 8.803004000521610256395093315913, 9.357825177103752954290361116768
