L(s) = 1 | + (1.16 − 0.803i)2-s + (0.709 − 1.87i)4-s + (1.29 + 1.82i)5-s + (−0.306 − 0.306i)7-s + (−0.676 − 2.74i)8-s + (2.97 + 1.08i)10-s + 4.06·11-s + (0.625 + 0.625i)13-s + (−0.603 − 0.110i)14-s + (−2.99 − 2.65i)16-s + (3.57 − 3.57i)17-s − 6.82·19-s + (4.32 − 1.12i)20-s + (4.73 − 3.26i)22-s + (−1.58 − 1.58i)23-s + ⋯ |
L(s) = 1 | + (0.822 − 0.568i)2-s + (0.354 − 0.935i)4-s + (0.578 + 0.815i)5-s + (−0.115 − 0.115i)7-s + (−0.239 − 0.970i)8-s + (0.939 + 0.343i)10-s + 1.22·11-s + (0.173 + 0.173i)13-s + (−0.161 − 0.0295i)14-s + (−0.748 − 0.663i)16-s + (0.867 − 0.867i)17-s − 1.56·19-s + (0.967 − 0.251i)20-s + (1.00 − 0.696i)22-s + (−0.331 − 0.331i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.12954 - 0.868779i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.12954 - 0.868779i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.16 + 0.803i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.29 - 1.82i)T \) |
good | 7 | \( 1 + (0.306 + 0.306i)T + 7iT^{2} \) |
| 11 | \( 1 - 4.06T + 11T^{2} \) |
| 13 | \( 1 + (-0.625 - 0.625i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.57 + 3.57i)T - 17iT^{2} \) |
| 19 | \( 1 + 6.82T + 19T^{2} \) |
| 23 | \( 1 + (1.58 + 1.58i)T + 23iT^{2} \) |
| 29 | \( 1 - 8.50iT - 29T^{2} \) |
| 31 | \( 1 + 2.56T + 31T^{2} \) |
| 37 | \( 1 + (1.69 - 1.69i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.19iT - 41T^{2} \) |
| 43 | \( 1 + (4.18 + 4.18i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.58 + 4.58i)T - 47iT^{2} \) |
| 53 | \( 1 + (7.41 - 7.41i)T - 53iT^{2} \) |
| 59 | \( 1 + 8.79iT - 59T^{2} \) |
| 61 | \( 1 - 6.08iT - 61T^{2} \) |
| 67 | \( 1 + (6.18 - 6.18i)T - 67iT^{2} \) |
| 71 | \( 1 - 14.7iT - 71T^{2} \) |
| 73 | \( 1 + (-3.05 + 3.05i)T - 73iT^{2} \) |
| 79 | \( 1 - 8.56iT - 79T^{2} \) |
| 83 | \( 1 + (-5.13 + 5.13i)T - 83iT^{2} \) |
| 89 | \( 1 + 9.88T + 89T^{2} \) |
| 97 | \( 1 + (-1.75 - 1.75i)T + 97iT^{2} \) |
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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
−11.36325902809832438552868187363, −10.54721740870818297897317477754, −9.795070786956172998198976267243, −8.829070920154491527971046234498, −7.03294920235932742771914517906, −6.47909080380202478823359351746, −5.42070124972916454279330331641, −4.08754248319983243902105627159, −3.05718981795471473876333546104, −1.68933124662226169449339701436,
1.92067973328556760878445612208, 3.70523777322112034633367453714, 4.59495089096359839840430377575, 5.93173424048146827938613971620, 6.31493035079758313329779104534, 7.79386839102462578610637264810, 8.637607135352974085695573493842, 9.515466176667216218470500872223, 10.77026746729583205865956039527, 11.96956273574504291504166356948