| L(s) = 1 | + (−1.12 + 1.65i)2-s + (0.163 − 2.99i)3-s + (−1.45 − 3.72i)4-s + (2.22 − 4.47i)5-s + (4.76 + 3.64i)6-s + (3.56 − 3.56i)7-s + (7.79 + 1.79i)8-s + (−8.94 − 0.978i)9-s + (4.88 + 8.72i)10-s + (7.74 + 5.62i)11-s + (−11.3 + 3.75i)12-s + (2.47 − 15.6i)13-s + (1.86 + 9.92i)14-s + (−13.0 − 7.39i)15-s + (−11.7 + 10.8i)16-s + (4.02 + 7.90i)17-s + ⋯ |
| L(s) = 1 | + (−0.563 + 0.825i)2-s + (0.0544 − 0.998i)3-s + (−0.364 − 0.931i)4-s + (0.444 − 0.895i)5-s + (0.793 + 0.608i)6-s + (0.509 − 0.509i)7-s + (0.974 + 0.224i)8-s + (−0.994 − 0.108i)9-s + (0.488 + 0.872i)10-s + (0.704 + 0.511i)11-s + (−0.949 + 0.312i)12-s + (0.190 − 1.20i)13-s + (0.133 + 0.708i)14-s + (−0.869 − 0.493i)15-s + (−0.734 + 0.678i)16-s + (0.236 + 0.464i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.808732 - 0.895314i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.808732 - 0.895314i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.12 - 1.65i)T \) |
| 3 | \( 1 + (-0.163 + 2.99i)T \) |
| 5 | \( 1 + (-2.22 + 4.47i)T \) |
| good | 7 | \( 1 + (-3.56 + 3.56i)T - 49iT^{2} \) |
| 11 | \( 1 + (-7.74 - 5.62i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (-2.47 + 15.6i)T + (-160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (-4.02 - 7.90i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (-0.403 + 1.24i)T + (-292. - 212. i)T^{2} \) |
| 23 | \( 1 + (27.0 - 4.28i)T + (503. - 163. i)T^{2} \) |
| 29 | \( 1 + (17.6 + 54.4i)T + (-680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-0.688 - 0.223i)T + (777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-30.8 - 4.88i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (18.5 + 25.4i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-47.3 - 47.3i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (14.1 - 27.7i)T + (-1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-10.4 + 20.4i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (3.62 + 4.99i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (58.2 + 42.3i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (15.7 + 30.8i)T + (-2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (-20.4 - 62.9i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-89.9 + 14.2i)T + (5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (32.1 + 98.9i)T + (-5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (5.27 + 10.3i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (87.4 + 63.5i)T + (2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (49.4 - 97.0i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
| 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
−11.25298195362541277013907498364, −10.04715480150989073894330002517, −9.192825352714547162642925077748, −7.952452712487287801216138348648, −7.82176170045438633065829016344, −6.30872613231923017250710750322, −5.66587924124445928733688772358, −4.33201948522409738850325503795, −1.84900767707405507113212655325, −0.73082785604808792086182637326,
1.96390314926738263045667424232, 3.23974291922461166336517477486, 4.25746940694727564667813079935, 5.67446215894363523656737349570, 7.01690495708008921367243003638, 8.426824238050502345811224349369, 9.184005478314929328300706868097, 9.884516065821307020391728185912, 10.91097903022051816488747965963, 11.40268936281906106670621096261
