L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (2.63 + 0.189i)7-s + 0.999i·8-s + (4.67 − 2.69i)11-s − 2.51i·13-s + (2.19 + 1.48i)14-s + (−0.5 + 0.866i)16-s + (2.24 + 3.89i)17-s + (−2.48 − 1.43i)19-s + 5.39·22-s + (0.232 + 0.133i)23-s + (1.25 − 2.18i)26-s + (1.15 + 2.38i)28-s − 8.89i·29-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.997 + 0.0716i)7-s + 0.353i·8-s + (1.40 − 0.813i)11-s − 0.698i·13-s + (0.585 + 0.396i)14-s + (−0.125 + 0.216i)16-s + (0.545 + 0.945i)17-s + (−0.568 − 0.328i)19-s + 1.15·22-s + (0.0483 + 0.0279i)23-s + (0.246 − 0.427i)26-s + (0.218 + 0.449i)28-s − 1.65i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.410197034\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.410197034\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.63 - 0.189i)T \) |
good | 11 | \( 1 + (-4.67 + 2.69i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2.51iT - 13T^{2} \) |
| 17 | \( 1 + (-2.24 - 3.89i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.48 + 1.43i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.232 - 0.133i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 8.89iT - 29T^{2} \) |
| 31 | \( 1 + (-4.18 + 2.41i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.25 - 5.64i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 0.760T + 41T^{2} \) |
| 43 | \( 1 - 5.86T + 43T^{2} \) |
| 47 | \( 1 + (-3.99 + 6.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7.27 - 4.19i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6.33 + 10.9i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.27 + 1.31i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.91 - 8.50i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4.76iT - 71T^{2} \) |
| 73 | \( 1 + (10.0 - 5.82i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.29 - 7.44i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 9.45T + 83T^{2} \) |
| 89 | \( 1 + (-3.98 + 6.90i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6.16iT - 97T^{2} \) |
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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
−8.344440947451061409186346705978, −8.145531878105953357175562832905, −7.15877360326066926881171441547, −6.16133314356926365306737001404, −5.87878381006405849963267041398, −4.78802794622592634822674047456, −4.10475274433088590661407746465, −3.33402712191915878305625853260, −2.15986903264985027112467001917, −1.01768218163845986120464682904,
1.25077375353777933346491739195, 1.90723850324376831958447251358, 3.09837895885299246185249207642, 4.15826323398127330277547961002, 4.59934400997275914258241498441, 5.41978160574850967261949034324, 6.40777888761853883387238004928, 7.07921705520863475918737097274, 7.73213356158928520438601144323, 8.948313272733850174634225295022