| L(s) = 1 | + (−0.951 + 0.309i)2-s + (−1.35 − 1.07i)3-s + (0.809 − 0.587i)4-s + (0.866 − 0.5i)5-s + (1.62 + 0.603i)6-s + (0.0557 + 0.530i)7-s + (−0.587 + 0.809i)8-s + (0.685 + 2.92i)9-s + (−0.669 + 0.743i)10-s + (−3.26 − 1.45i)11-s + (−1.73 − 0.0724i)12-s + (−0.192 − 0.903i)13-s + (−0.217 − 0.487i)14-s + (−1.71 − 0.252i)15-s + (0.309 − 0.951i)16-s + (4.44 − 1.98i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.218i)2-s + (−0.783 − 0.621i)3-s + (0.404 − 0.293i)4-s + (0.387 − 0.223i)5-s + (0.662 + 0.246i)6-s + (0.0210 + 0.200i)7-s + (−0.207 + 0.286i)8-s + (0.228 + 0.973i)9-s + (−0.211 + 0.235i)10-s + (−0.985 − 0.438i)11-s + (−0.499 − 0.0209i)12-s + (−0.0532 − 0.250i)13-s + (−0.0580 − 0.130i)14-s + (−0.442 − 0.0653i)15-s + (0.0772 − 0.237i)16-s + (1.07 − 0.480i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.131 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.131 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.478442 - 0.545974i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.478442 - 0.545974i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 3 | \( 1 + (1.35 + 1.07i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 31 | \( 1 + (2.08 - 5.16i)T \) |
| good | 7 | \( 1 + (-0.0557 - 0.530i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (3.26 + 1.45i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (0.192 + 0.903i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-4.44 + 1.98i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-6.98 - 1.48i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (2.96 + 2.15i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.61 + 4.97i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (6.62 + 3.82i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.30 - 6.57i)T + (4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-0.884 + 4.16i)T + (-39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (10.3 + 3.36i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.214 + 2.04i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-5.93 + 5.34i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + 2.90iT - 61T^{2} \) |
| 67 | \( 1 + (7.79 + 13.5i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (10.3 + 1.09i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (-6.67 + 14.9i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (3.95 + 8.88i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (2.52 - 2.80i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (-1.09 + 0.796i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-6.70 + 4.87i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.01810633077533128817782949883, −8.990899369084043668291054331006, −7.82719669275066289522177764439, −7.60065683037827147128326834721, −6.37312403045818786333178799734, −5.54491334659886097756383673148, −5.08909833146127313179931917803, −3.13079918027216464353786082702, −1.83612032009917455262383697845, −0.51863918397535346864365326943,
1.28717449079283596221708691869, 2.87045876316713473951265567015, 3.95164008118825852246857909305, 5.27104202234133764187240685046, 5.80650245538761165189920696427, 7.09251548771662738854850021457, 7.64226850623728185463424939174, 8.885455419911095800180765565805, 9.848870521628974218980745496815, 10.07120863750979884333495773907
