| L(s) = 1 | + (0.5 + 0.133i)2-s + (−0.366 − 0.633i)3-s + (−3.23 − 1.86i)4-s + (−0.0980 − 0.366i)6-s + (−5.73 + 1.53i)7-s + (−2.83 − 2.83i)8-s + (4.23 − 7.33i)9-s + (−4.19 + 15.6i)11-s + 2.73i·12-s + (6.5 + 11.2i)13-s − 3.07·14-s + (6.42 + 11.1i)16-s + (15.9 + 9.23i)17-s + (3.09 − 3.09i)18-s + (1.63 + 6.09i)19-s + ⋯ |
| L(s) = 1 | + (0.250 + 0.0669i)2-s + (−0.122 − 0.211i)3-s + (−0.808 − 0.466i)4-s + (−0.0163 − 0.0610i)6-s + (−0.818 + 0.219i)7-s + (−0.353 − 0.353i)8-s + (0.470 − 0.814i)9-s + (−0.381 + 1.42i)11-s + 0.227i·12-s + (0.5 + 0.866i)13-s − 0.219·14-s + (0.401 + 0.695i)16-s + (0.940 + 0.543i)17-s + (0.172 − 0.172i)18-s + (0.0859 + 0.320i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0257 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0257 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.615275 + 0.599609i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.615275 + 0.599609i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + (-6.5 - 11.2i)T \) |
| good | 2 | \( 1 + (-0.5 - 0.133i)T + (3.46 + 2i)T^{2} \) |
| 3 | \( 1 + (0.366 + 0.633i)T + (-4.5 + 7.79i)T^{2} \) |
| 7 | \( 1 + (5.73 - 1.53i)T + (42.4 - 24.5i)T^{2} \) |
| 11 | \( 1 + (4.19 - 15.6i)T + (-104. - 60.5i)T^{2} \) |
| 17 | \( 1 + (-15.9 - 9.23i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-1.63 - 6.09i)T + (-312. + 180.5i)T^{2} \) |
| 23 | \( 1 + (17.4 - 10.0i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (4.69 + 8.13i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (11.9 - 11.9i)T - 961iT^{2} \) |
| 37 | \( 1 + (8.11 - 30.2i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + (-44.9 - 12.0i)T + (1.45e3 + 840.5i)T^{2} \) |
| 43 | \( 1 + (45 + 25.9i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-34.3 - 34.3i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 - 14.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + (92.9 - 24.9i)T + (3.01e3 - 1.74e3i)T^{2} \) |
| 61 | \( 1 + (12.8 - 22.1i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (39.0 + 10.4i)T + (3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + (11.9 + 44.6i)T + (-4.36e3 + 2.52e3i)T^{2} \) |
| 73 | \( 1 + (-19.2 - 19.2i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 62.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-24.4 + 24.4i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (23.1 - 86.4i)T + (-6.85e3 - 3.96e3i)T^{2} \) |
| 97 | \( 1 + (14.1 + 52.9i)T + (-8.14e3 + 4.70e3i)T^{2} \) |
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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
−12.03380202572615573568323187622, −10.39032570949368048851795924432, −9.694771105572760304951950548272, −9.127269767578771446927771119482, −7.71898875282326866418493364040, −6.56617578496520108333253429649, −5.80914148037364912617833591871, −4.48931154238928254744627582919, −3.55585179650034429608164199859, −1.51613323412008809140422631427,
0.40745456753584583942618976165, 2.97733917167772075836073643147, 3.82440007241116884110295603572, 5.16373388566569439269222677677, 5.95386660360421662906787096989, 7.54913128864956785521584585635, 8.275130431564399386939725183347, 9.329904732241897497222861731567, 10.27203186991268143203482505435, 11.06319478748102996187285025849
