L(s) = 1 | + (1 + 1.41i)3-s − 2·5-s + (−0.949 + 0.548i)7-s + (−1.00 + 2.82i)9-s + (0.724 + 1.25i)11-s + (4.5 + 2.59i)13-s + (−2 − 2.82i)15-s + (−5.17 − 2.98i)17-s + (−1.5 + 2.59i)19-s + (−1.72 − 0.794i)21-s + (−0.825 − 0.476i)23-s − 25-s + (−5.00 + 1.41i)27-s + (−4.62 + 2.66i)29-s + (−1.5 + 0.866i)31-s + ⋯ |
L(s) = 1 | + (0.577 + 0.816i)3-s − 0.894·5-s + (−0.358 + 0.207i)7-s + (−0.333 + 0.942i)9-s + (0.218 + 0.378i)11-s + (1.24 + 0.720i)13-s + (−0.516 − 0.730i)15-s + (−1.25 − 0.724i)17-s + (−0.344 + 0.596i)19-s + (−0.376 − 0.173i)21-s + (−0.172 − 0.0994i)23-s − 0.200·25-s + (−0.962 + 0.272i)27-s + (−0.858 + 0.495i)29-s + (−0.269 + 0.155i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 - 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.273049 + 0.995930i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.273049 + 0.995930i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1 - 1.41i)T \) |
| 67 | \( 1 + (-8 + 1.73i)T \) |
good | 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 + (0.949 - 0.548i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.724 - 1.25i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.5 - 2.59i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (5.17 + 2.98i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 - 2.59i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.825 + 0.476i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.62 - 2.66i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.94 - 8.57i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.72 + 2.98i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + 1.27iT - 43T^{2} \) |
| 47 | \( 1 + (0.275 - 0.158i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 0.635iT - 59T^{2} \) |
| 61 | \( 1 + (-9.39 - 5.42i)T + (30.5 + 52.8i)T^{2} \) |
| 71 | \( 1 + (-11.1 + 6.45i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.94 + 5.10i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.398 + 0.230i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (10.0 + 5.81i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 4.73iT - 89T^{2} \) |
| 97 | \( 1 + (-11.8 - 6.84i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.64170164139124281138222471733, −9.628230741226306898101211077296, −8.876240615322327879161021488910, −8.328498121704622611852105630872, −7.25410243528187087338243911908, −6.32364909425578328219476656322, −5.01761970851814346606948405147, −4.04683176555877781936001640893, −3.48618449586255284097732342985, −2.05437118195364086643866712957,
0.46308863549140384750307188062, 2.08165071876625340917275967084, 3.51374106422397149560284641196, 3.98759498332930193505150608106, 5.72313604307986687771539784995, 6.58383774324913178166195040662, 7.34791836989052170388315673152, 8.395379907763502545382756816553, 8.623494072389611927595989610770, 9.791931121141788016153070242392