L(s) = 1 | + (0.939 − 0.342i)3-s + (−0.749 + 4.25i)5-s + (−3.63 + 2.09i)7-s + (0.766 − 0.642i)9-s + (−3.21 − 1.85i)11-s + (1.64 − 4.52i)13-s + (0.749 + 4.25i)15-s + (0.869 + 0.729i)17-s + (−4.06 + 1.57i)19-s + (−2.69 + 3.21i)21-s + (0.795 − 0.140i)23-s + (−12.8 − 4.66i)25-s + (0.500 − 0.866i)27-s + (2.83 + 3.37i)29-s + (−2.66 − 4.61i)31-s + ⋯ |
L(s) = 1 | + (0.542 − 0.197i)3-s + (−0.335 + 1.90i)5-s + (−1.37 + 0.792i)7-s + (0.255 − 0.214i)9-s + (−0.968 − 0.559i)11-s + (0.457 − 1.25i)13-s + (0.193 + 1.09i)15-s + (0.210 + 0.176i)17-s + (−0.932 + 0.361i)19-s + (−0.588 + 0.700i)21-s + (0.165 − 0.0292i)23-s + (−2.56 − 0.933i)25-s + (0.0962 − 0.166i)27-s + (0.526 + 0.627i)29-s + (−0.478 − 0.829i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0209i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00557055 - 0.530504i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00557055 - 0.530504i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.939 + 0.342i)T \) |
| 19 | \( 1 + (4.06 - 1.57i)T \) |
good | 5 | \( 1 + (0.749 - 4.25i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (3.63 - 2.09i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.21 + 1.85i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.64 + 4.52i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.869 - 0.729i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.795 + 0.140i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.83 - 3.37i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (2.66 + 4.61i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.90iT - 37T^{2} \) |
| 41 | \( 1 + (1.34 + 3.69i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (4.88 + 0.860i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.921 - 1.09i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (9.28 - 1.63i)T + (49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-1.86 - 1.56i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-2.46 - 13.9i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (0.583 - 0.489i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.17 - 12.3i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (7.65 - 2.78i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (5.89 - 2.14i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-5.90 + 3.40i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.639 - 1.75i)T + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (10.5 - 12.5i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−10.35051999200299620839418885848, −9.992789729006685124654951243058, −8.693519967010836660717144417387, −7.968830271468409222324221419435, −7.09511963709262738387040071516, −6.26452043125430193386051965295, −5.69985071962474105515471133662, −3.70403043990400364263286035753, −3.02387903717406894075678338598, −2.54370175392310692467677788682,
0.21930207734838549589881648597, 1.80340510327240565857055078674, 3.40406604579429956863598647546, 4.34478522568180950149467353118, 4.91224534543735000608702298516, 6.25968645615168708586968835740, 7.24737905810465240964445242738, 8.160707899507185540082712939678, 8.927599909911147060649705736500, 9.524634168200604045989240779550