L(s) = 1 | − 1.20·2-s + (−1.24 + 2.15i)3-s − 0.547·4-s + (0.667 + 1.15i)5-s + (1.50 − 2.59i)6-s + (1.94 − 3.36i)7-s + 3.07·8-s + (−1.60 − 2.77i)9-s + (−0.804 − 1.39i)10-s + (−2.37 − 4.11i)11-s + (0.681 − 1.18i)12-s + (0.5 + 0.866i)13-s + (−2.34 + 4.05i)14-s − 3.32·15-s − 2.60·16-s + (2.91 − 5.05i)17-s + ⋯ |
L(s) = 1 | − 0.852·2-s + (−0.718 + 1.24i)3-s − 0.273·4-s + (0.298 + 0.516i)5-s + (0.612 − 1.06i)6-s + (0.734 − 1.27i)7-s + 1.08·8-s + (−0.533 − 0.924i)9-s + (−0.254 − 0.440i)10-s + (−0.715 − 1.23i)11-s + (0.196 − 0.340i)12-s + (0.138 + 0.240i)13-s + (−0.626 + 1.08i)14-s − 0.857·15-s − 0.651·16-s + (0.708 − 1.22i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.660422 + 0.0821717i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.660422 + 0.0821717i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (-5.13 - 2.16i)T \) |
good | 2 | \( 1 + 1.20T + 2T^{2} \) |
| 3 | \( 1 + (1.24 - 2.15i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.667 - 1.15i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.94 + 3.36i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.37 + 4.11i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.91 + 5.05i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.16 - 3.75i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 7.18T + 23T^{2} \) |
| 29 | \( 1 - 4.46T + 29T^{2} \) |
| 37 | \( 1 + (2.02 - 3.51i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.472 - 0.819i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.744 + 1.28i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + (-1.30 - 2.25i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.07 + 12.2i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 - 1.94T + 61T^{2} \) |
| 67 | \( 1 + (-2.95 - 5.12i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.68 + 4.65i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.21 - 12.4i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.96 + 5.13i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7.47 + 12.9i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 - 6.62T + 97T^{2} \) |
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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
−10.89174721683921317152158192598, −10.32233919259970194200605034547, −9.875008709638722816173410756362, −8.641737113050868135775712690344, −7.83880941780476501756135808042, −6.65773394634693843283568574770, −5.19961183377777687190786316359, −4.60222569748655740122311693229, −3.33793518411463022614368150930, −0.814440511479119700525556088772,
1.18026291418277686579373951181, 2.20947453767380637945530106080, 4.83342707837607087198099693482, 5.41050127466317831784071479519, 6.67534877076815526603773346224, 7.71671330912339658534956149866, 8.407295699031298812947559975491, 9.200184201172269165785493285890, 10.29860318262720268197766629242, 11.23590724283952276038509866189