L(s) = 1 | + (−0.965 + 0.258i)2-s + (−0.523 + 1.95i)3-s + (0.866 − 0.499i)4-s + (2.03 + 0.935i)5-s − 2.02i·6-s + (−0.707 + 0.707i)8-s + (−0.941 − 0.543i)9-s + (−2.20 − 0.378i)10-s + (2.01 + 3.49i)11-s + (0.523 + 1.95i)12-s + (−0.204 − 0.204i)13-s + (−2.89 + 3.47i)15-s + (0.500 − 0.866i)16-s + (1.97 + 0.527i)17-s + (1.05 + 0.281i)18-s + (3.10 − 5.37i)19-s + ⋯ |
L(s) = 1 | + (−0.683 + 0.183i)2-s + (−0.302 + 1.12i)3-s + (0.433 − 0.249i)4-s + (0.908 + 0.418i)5-s − 0.825i·6-s + (−0.249 + 0.249i)8-s + (−0.313 − 0.181i)9-s + (−0.696 − 0.119i)10-s + (0.609 + 1.05i)11-s + (0.151 + 0.563i)12-s + (−0.0568 − 0.0568i)13-s + (−0.746 + 0.897i)15-s + (0.125 − 0.216i)16-s + (0.477 + 0.128i)17-s + (0.247 + 0.0663i)18-s + (0.711 − 1.23i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.432 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.599739 + 0.952473i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.599739 + 0.952473i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 - 0.258i)T \) |
| 5 | \( 1 + (-2.03 - 0.935i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.523 - 1.95i)T + (-2.59 - 1.5i)T^{2} \) |
| 11 | \( 1 + (-2.01 - 3.49i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.204 + 0.204i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.97 - 0.527i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-3.10 + 5.37i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.17 - 4.38i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 7.15iT - 29T^{2} \) |
| 31 | \( 1 + (6.33 - 3.65i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.46 - 1.19i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 2.58iT - 41T^{2} \) |
| 43 | \( 1 + (4.97 - 4.97i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.0815 - 0.304i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (8.00 + 2.14i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (0.427 + 0.740i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.99 - 3.46i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.817 - 3.05i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 7.12T + 71T^{2} \) |
| 73 | \( 1 + (-2.98 + 11.1i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (4.39 + 2.53i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.85 - 3.85i)T + 83iT^{2} \) |
| 89 | \( 1 + (-1.53 + 2.66i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.63 + 6.63i)T - 97iT^{2} \) |
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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
−10.97668418187131103095958674537, −10.07231057097247276627203380274, −9.623328128419040690090482309689, −9.038653102865992559295282598128, −7.51577156441417726302358689501, −6.74404640934934640119030462933, −5.57128453618903772162118974788, −4.76791271784823037204880699179, −3.33860886762191192477045413286, −1.78502806826226696967028739713,
0.962984713459045393296758331684, 1.90133665664182906792093597795, 3.46895599709752091663995666788, 5.37779223386916490802166044647, 6.20450890981575508840799892430, 6.99058635199250325969584460597, 8.030976109323944964707211356295, 8.866843581607155956405737915325, 9.714177743037153723021655116615, 10.64646236090463054473380262042