L(s) = 1 | + (−1.28 − 3.78i)2-s + 5.19i·3-s + (−12.7 + 9.70i)4-s + (19.6 − 6.65i)6-s − 89.0i·7-s + (53.0 + 35.7i)8-s − 27·9-s + 174. i·11-s + (−50.4 − 66.1i)12-s + 22.9·13-s + (−337. + 114. i)14-s + (67.7 − 246. i)16-s − 69.2·17-s + (34.5 + 102. i)18-s − 341. i·19-s + ⋯ |
L(s) = 1 | + (−0.320 − 0.947i)2-s + 0.577i·3-s + (−0.795 + 0.606i)4-s + (0.546 − 0.184i)6-s − 1.81i·7-s + (0.828 + 0.559i)8-s − 0.333·9-s + 1.44i·11-s + (−0.350 − 0.459i)12-s + 0.136·13-s + (−1.72 + 0.581i)14-s + (0.264 − 0.964i)16-s − 0.239·17-s + (0.106 + 0.315i)18-s − 0.944i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 - 0.795i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.02276098926\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02276098926\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.28 + 3.78i)T \) |
| 3 | \( 1 - 5.19iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 89.0iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 174. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 22.9T + 2.85e4T^{2} \) |
| 17 | \( 1 + 69.2T + 8.35e4T^{2} \) |
| 19 | \( 1 + 341. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 319. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 679.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 72.5iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 2.37e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 762.T + 2.82e6T^{2} \) |
| 43 | \( 1 - 3.11e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 315. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.38e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 6.68e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 5.31e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 4.01e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 2.95e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 5.74e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 414. iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 9.73e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 8.19e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 9.56e3T + 8.85e7T^{2} \) |
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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
−10.40631415927259081853367264510, −9.915589647604237132233827221515, −8.887588049488343043382782173793, −7.67560243739072656047020994823, −6.84367830268200173486926615611, −4.68430914913902880030871992804, −4.32619074219774044066545618099, −3.02150057974681659085221160655, −1.41860028081020085546024184171, −0.008096590414680710627243170113,
1.69814727461648609327160061324, 3.31601863510422575802153632640, 5.26326821367428326124826658151, 5.86201201785906956858390953375, 6.69964662612869379051768490993, 8.149098344286388231474535165524, 8.564099995862248642075635553031, 9.400827938791294510811960141473, 10.72615511587502281537102486714, 11.84838161466726236441338310056