L(s) = 1 | + (1.66 − 1.66i)3-s − 1.87i·7-s − 2.56i·9-s + (3.29 + 3.29i)11-s + (−1.90 + 1.90i)13-s − 2.57·17-s + (5.76 − 5.76i)19-s + (−3.12 − 3.12i)21-s − 7.58i·23-s + (0.728 + 0.728i)27-s + (6.45 − 6.45i)29-s + 0.799·31-s + 10.9·33-s + (−2.69 − 2.69i)37-s + 6.33i·39-s + ⋯ |
L(s) = 1 | + (0.962 − 0.962i)3-s − 0.708i·7-s − 0.854i·9-s + (0.994 + 0.994i)11-s + (−0.527 + 0.527i)13-s − 0.623·17-s + (1.32 − 1.32i)19-s + (−0.681 − 0.681i)21-s − 1.58i·23-s + (0.140 + 0.140i)27-s + (1.19 − 1.19i)29-s + 0.143·31-s + 1.91·33-s + (−0.443 − 0.443i)37-s + 1.01i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.467969551\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.467969551\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-1.66 + 1.66i)T - 3iT^{2} \) |
| 7 | \( 1 + 1.87iT - 7T^{2} \) |
| 11 | \( 1 + (-3.29 - 3.29i)T + 11iT^{2} \) |
| 13 | \( 1 + (1.90 - 1.90i)T - 13iT^{2} \) |
| 17 | \( 1 + 2.57T + 17T^{2} \) |
| 19 | \( 1 + (-5.76 + 5.76i)T - 19iT^{2} \) |
| 23 | \( 1 + 7.58iT - 23T^{2} \) |
| 29 | \( 1 + (-6.45 + 6.45i)T - 29iT^{2} \) |
| 31 | \( 1 - 0.799T + 31T^{2} \) |
| 37 | \( 1 + (2.69 + 2.69i)T + 37iT^{2} \) |
| 41 | \( 1 - 0.946iT - 41T^{2} \) |
| 43 | \( 1 + (-0.829 - 0.829i)T + 43iT^{2} \) |
| 47 | \( 1 - 1.52T + 47T^{2} \) |
| 53 | \( 1 + (6.97 + 6.97i)T + 53iT^{2} \) |
| 59 | \( 1 + (6.84 + 6.84i)T + 59iT^{2} \) |
| 61 | \( 1 + (6.87 - 6.87i)T - 61iT^{2} \) |
| 67 | \( 1 + (3.73 - 3.73i)T - 67iT^{2} \) |
| 71 | \( 1 - 9.34iT - 71T^{2} \) |
| 73 | \( 1 - 0.886iT - 73T^{2} \) |
| 79 | \( 1 + 3.07T + 79T^{2} \) |
| 83 | \( 1 + (0.989 - 0.989i)T - 83iT^{2} \) |
| 89 | \( 1 - 10.0iT - 89T^{2} \) |
| 97 | \( 1 - 7.16T + 97T^{2} \) |
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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
−9.161514318130248431641496626106, −8.391347933204344306997244852383, −7.48876898922610601783186757958, −6.94300112837868281532823720894, −6.47967533880924379350231957304, −4.79688303290712051422043031266, −4.20965160645400761621810502661, −2.87777468897016129294984629859, −2.10902096273311073653687389858, −0.926771245691294866595171530258,
1.50014047497346824061973168918, 3.09544831340505466699385633767, 3.32283000026959076520289228505, 4.46685288654345925147948393391, 5.44230680564298858348756509828, 6.19506197185819513662572757042, 7.42665230518755137822271885374, 8.237256294711537289504558796049, 8.991016244696454187864163900079, 9.374186941817776546326709653469