L(s) = 1 | + (−1.57 − 0.719i)3-s + (−3.36 + 2.91i)7-s + (1.96 + 2.26i)9-s + (1.63 − 2.54i)13-s + (4.23 − 4.89i)19-s + (7.39 − 2.17i)21-s + (−4.20 − 2.70i)25-s + (−1.46 − 4.98i)27-s + (−6.00 − 9.33i)31-s + 12.1·37-s + (−4.40 + 2.83i)39-s + (−12.9 + 1.85i)43-s + (1.82 − 12.6i)49-s + (−10.2 + 4.65i)57-s + (−4.39 − 14.9i)61-s + ⋯ |
L(s) = 1 | + (−0.909 − 0.415i)3-s + (−1.27 + 1.10i)7-s + (0.654 + 0.755i)9-s + (0.453 − 0.705i)13-s + (0.972 − 1.12i)19-s + (1.61 − 0.474i)21-s + (−0.841 − 0.540i)25-s + (−0.281 − 0.959i)27-s + (−1.07 − 1.67i)31-s + 1.99·37-s + (−0.705 + 0.453i)39-s + (−1.97 + 0.283i)43-s + (0.260 − 1.81i)49-s + (−1.35 + 0.617i)57-s + (−0.563 − 1.91i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.431835 - 0.488089i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.431835 - 0.488089i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.57 + 0.719i)T \) |
| 67 | \( 1 + (-5.79 - 5.78i)T \) |
good | 5 | \( 1 + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (3.36 - 2.91i)T + (0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.63 + 2.54i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-4.23 + 4.89i)T + (-2.70 - 18.8i)T^{2} \) |
| 23 | \( 1 + (15.0 + 17.3i)T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + (6.00 + 9.33i)T + (-12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 - 12.1T + 37T^{2} \) |
| 41 | \( 1 + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (12.9 - 1.85i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + (30.7 + 35.5i)T^{2} \) |
| 53 | \( 1 + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (4.39 + 14.9i)T + (-51.3 + 32.9i)T^{2} \) |
| 71 | \( 1 + (68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-5.46 + 1.60i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-9.04 + 14.0i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + 6.97iT - 97T^{2} \) |
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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
−9.841221091492245139526657776135, −9.484682569489210891643452231736, −8.242123284461205948597781820560, −7.33010946127707678816350449370, −6.24993565735497314702906765617, −5.87918702234907039733908890693, −4.87872879191395068358972557385, −3.41193702598138738284360745320, −2.27093230583196490948465701826, −0.40515711029627455510024832309,
1.24276202379162434812750458360, 3.44762618334476231942376883039, 3.96398172182685627390699410753, 5.21659051716985217165840584439, 6.19700225336367005655039930765, 6.84514953456152490999074587030, 7.68982946860865905980184811376, 9.146361670114100373876052145747, 9.823855832931944090465214437108, 10.37807484587014335816469859187