| L(s) = 1 | + (−0.970 − 1.02i)2-s + (1.04 − 1.37i)3-s + (−0.116 + 1.99i)4-s + (0.178 − 0.267i)5-s + (−2.43 + 0.260i)6-s + (−0.862 − 2.08i)7-s + (2.16 − 1.81i)8-s + (−0.803 − 2.89i)9-s + (−0.449 + 0.0758i)10-s + (−1.97 + 0.392i)11-s + (2.63 + 2.25i)12-s + (5.61 − 3.75i)13-s + (−1.30 + 2.90i)14-s + (−0.181 − 0.527i)15-s + (−3.97 − 0.464i)16-s + (−1.92 + 1.92i)17-s + ⋯ |
| L(s) = 1 | + (−0.686 − 0.727i)2-s + (0.605 − 0.796i)3-s + (−0.0581 + 0.998i)4-s + (0.0800 − 0.119i)5-s + (−0.994 + 0.106i)6-s + (−0.326 − 0.787i)7-s + (0.766 − 0.642i)8-s + (−0.267 − 0.963i)9-s + (−0.142 + 0.0239i)10-s + (−0.594 + 0.118i)11-s + (0.759 + 0.650i)12-s + (1.55 − 1.04i)13-s + (−0.348 + 0.777i)14-s + (−0.0469 − 0.136i)15-s + (−0.993 − 0.116i)16-s + (−0.467 + 0.467i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.481268 - 0.839614i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.481268 - 0.839614i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.970 + 1.02i)T \) |
| 3 | \( 1 + (-1.04 + 1.37i)T \) |
| good | 5 | \( 1 + (-0.178 + 0.267i)T + (-1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (0.862 + 2.08i)T + (-4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (1.97 - 0.392i)T + (10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (-5.61 + 3.75i)T + (4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (1.92 - 1.92i)T - 17iT^{2} \) |
| 19 | \( 1 + (3.54 - 2.36i)T + (7.27 - 17.5i)T^{2} \) |
| 23 | \( 1 + (-2.19 + 5.29i)T + (-16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (1.06 - 5.33i)T + (-26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 - 4.81T + 31T^{2} \) |
| 37 | \( 1 + (-1.42 + 2.13i)T + (-14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (-7.50 - 3.10i)T + (28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (-0.371 + 0.0738i)T + (39.7 - 16.4i)T^{2} \) |
| 47 | \( 1 + (-4.36 - 4.36i)T + 47iT^{2} \) |
| 53 | \( 1 + (-1.66 - 8.37i)T + (-48.9 + 20.2i)T^{2} \) |
| 59 | \( 1 + (-2.84 - 1.90i)T + (22.5 + 54.5i)T^{2} \) |
| 61 | \( 1 + (0.814 - 4.09i)T + (-56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (9.80 + 1.95i)T + (61.8 + 25.6i)T^{2} \) |
| 71 | \( 1 + (-4.67 + 1.93i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-3.32 - 1.37i)T + (51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (0.888 + 0.888i)T + 79iT^{2} \) |
| 83 | \( 1 + (7.24 + 10.8i)T + (-31.7 + 76.6i)T^{2} \) |
| 89 | \( 1 + (-6.70 + 2.77i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 - 17.3iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.48470651894260060106726853758, −10.91787666617169888074018117644, −10.46165197464592844060420997104, −9.025673020948415805798180107849, −8.318969929887382987712643160077, −7.40225346355121502478563940152, −6.23435746104867086424403324637, −3.99747254624856000202332527697, −2.82943786708072312360156369312, −1.11093264303286266287858257275,
2.42514149527015193539084794016, 4.28738610913146881119509339190, 5.62879653118923301593908408146, 6.67921685664052204851538184298, 8.152961399388265097510158752416, 8.883387046448799474949218450629, 9.554085901611456278449462321461, 10.69918866780147081722584732995, 11.45022858004690593456262566936, 13.33944174927408530301626995032
