| 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 + (0.137 + 2.64i)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 + (0.612 − 2.57i)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 + (0.0518 + 0.998i)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 + (0.163 − 0.687i)14-s + (−0.0210 + 0.0182i)15-s + (0.103 + 0.227i)16-s + (0.641 − 0.412i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.00125 + 0.760396i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00125 + 0.760396i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.137 - 2.64i)T \) |
| 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 | \( 1 + (-0.499 + 3.47i)T + (-93.0 - 27.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.05726834931077676227628040736, −9.090619093092204729194278021068, −8.928255255179229203663494539264, −7.76699301549096033852717555270, −7.15981718239931960443947198532, −5.71223828379770543382276526192, −5.16063623303931720272526278124, −3.46128614567005826554230072530, −2.88885292255859849521239030560, −1.47341066950750566535148417027,
0.73800333027602822187299141633, 2.05376568239407191793141443879, 3.32943465819038235066167517641, 4.50793782563224161891986377746, 5.66036665872137840456032511393, 6.85524608540933960899015268737, 7.43877647168797441457523018929, 7.954336747816080392069228741135, 9.187780587423862217728647681088, 9.574746548429874579588403533152
