L(s) = 1 | + (−0.959 − 0.281i)2-s + (−0.755 − 0.654i)3-s + (0.841 + 0.540i)4-s + (0.0153 − 0.106i)5-s + (0.540 + 0.841i)6-s + (−1.90 − 1.83i)7-s + (−0.654 − 0.755i)8-s + (0.142 + 0.989i)9-s + (−0.0447 + 0.0980i)10-s + (−0.605 − 2.06i)11-s + (−0.281 − 0.959i)12-s + (0.737 + 0.337i)13-s + (1.31 + 2.29i)14-s + (−0.0814 + 0.0706i)15-s + (0.415 + 0.909i)16-s + (−2.64 + 1.69i)17-s + ⋯ |
L(s) = 1 | + (−0.678 − 0.199i)2-s + (−0.436 − 0.378i)3-s + (0.420 + 0.270i)4-s + (0.00686 − 0.0477i)5-s + (0.220 + 0.343i)6-s + (−0.720 − 0.693i)7-s + (−0.231 − 0.267i)8-s + (0.0474 + 0.329i)9-s + (−0.0141 + 0.0310i)10-s + (−0.182 − 0.622i)11-s + (−0.0813 − 0.276i)12-s + (0.204 + 0.0934i)13-s + (0.350 + 0.613i)14-s + (−0.0210 + 0.0182i)15-s + (0.103 + 0.227i)16-s + (−0.641 + 0.412i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.462 - 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0417147 + 0.0688391i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0417147 + 0.0688391i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.959 + 0.281i)T \) |
| 3 | \( 1 + (0.755 + 0.654i)T \) |
| 7 | \( 1 + (1.90 + 1.83i)T \) |
| 23 | \( 1 + (-3.88 - 2.81i)T \) |
good | 5 | \( 1 + (-0.0153 + 0.106i)T + (-4.79 - 1.40i)T^{2} \) |
| 11 | \( 1 + (0.605 + 2.06i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-0.737 - 0.337i)T + (8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (2.64 - 1.69i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (4.49 + 2.89i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (7.56 - 4.85i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-5.87 + 5.09i)T + (4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (6.91 - 0.994i)T + (35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (10.8 + 1.55i)T + (39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-3.89 - 3.37i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 5.19iT - 47T^{2} \) |
| 53 | \( 1 + (5.98 - 2.73i)T + (34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (1.18 + 0.542i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (0.503 + 0.581i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (3.01 - 10.2i)T + (-56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-1.55 - 0.456i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-4.28 + 6.66i)T + (-30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-0.0576 - 0.0263i)T + (51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.979 - 6.81i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (9.03 - 10.4i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (0.499 - 3.47i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−10.54860511600915822033734945158, −9.369375086432181682065676874147, −8.751075034224436082550916733602, −7.77428349213359936560934286468, −6.83275008046217170981839328164, −6.39290294306495692458405517920, −5.15896649088155441408362505556, −3.88951585529864326103666946927, −2.80330336638128126185585280796, −1.33127842979045678750476092628,
0.05052769196461303648772317034, 2.01775067720768843783217431617, 3.21993852890799012162683199441, 4.58250578147499385304934429351, 5.51983141914160311069237202445, 6.51429292296657884234674039458, 6.98858780018116735668721151997, 8.357849051641356236983971374865, 8.907690950801559167064303569355, 9.786898182189972357160068384675