L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−2.09 − 1.62i)7-s + 0.999·8-s + (−2.59 + 1.5i)11-s + 2.44·13-s + (2.44 − 0.999i)14-s + (−0.5 + 0.866i)16-s + (0.878 − 0.507i)17-s + (0.878 + 0.507i)19-s − 3i·22-s + (−2.12 + 3.67i)23-s + (−1.22 + 2.12i)26-s + (−0.358 + 2.62i)28-s − 1.24i·29-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.790 − 0.612i)7-s + 0.353·8-s + (−0.783 + 0.452i)11-s + 0.679·13-s + (0.654 − 0.267i)14-s + (−0.125 + 0.216i)16-s + (0.213 − 0.123i)17-s + (0.201 + 0.116i)19-s − 0.639i·22-s + (−0.442 + 0.766i)23-s + (−0.240 + 0.416i)26-s + (−0.0677 + 0.495i)28-s − 0.230i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 - 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.683 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6803508220\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6803508220\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.09 + 1.62i)T \) |
good | 11 | \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 + (-0.878 + 0.507i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.878 - 0.507i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.12 - 3.67i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 1.24iT - 29T^{2} \) |
| 31 | \( 1 + (-4.86 + 2.80i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (7.13 + 4.12i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2.02T + 41T^{2} \) |
| 43 | \( 1 + 8.24iT - 43T^{2} \) |
| 47 | \( 1 + (0.878 + 0.507i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.621 + 1.07i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.76 - 9.98i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.12 - 2.95i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (8.66 - 5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.2iT - 71T^{2} \) |
| 73 | \( 1 + (4.18 + 7.24i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.62 - 9.73i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 3.16iT - 83T^{2} \) |
| 89 | \( 1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 3.76T + 97T^{2} \) |
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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
−8.874338256019644758105396905024, −8.147322351016893224892839188906, −7.35472027298648676042379777351, −6.92153347556923790070989043330, −5.92806897288792036647143982255, −5.41006979103571987066039643275, −4.28096335719037003699053984568, −3.55933923881666819745495990557, −2.39618119285263659760705717789, −1.03274135121225810912700855272,
0.27649738723943874449910614904, 1.65896241700584935050024241503, 2.85621654618056608942195261847, 3.26641393561378319125480958864, 4.40785975055105999973579497048, 5.35538887977348708506305178677, 6.16633415714141949806262534283, 6.86006648055251057387102711450, 7.987689578278984871137929785102, 8.464750860689275932695883654652