| L(s) = 1 | + (−1.30 − 2.25i)3-s + i·5-s + (−4.29 − 2.48i)7-s + (−1.89 + 3.27i)9-s + (−2.16 + 1.25i)11-s + (−1.78 − 3.13i)13-s + (2.25 − 1.30i)15-s + (0.505 − 0.874i)17-s + (4.03 + 2.32i)19-s + 12.9i·21-s + (3.06 + 5.31i)23-s − 25-s + 2.03·27-s + (0.890 + 1.54i)29-s − 1.65i·31-s + ⋯ |
| L(s) = 1 | + (−0.751 − 1.30i)3-s + 0.447i·5-s + (−1.62 − 0.937i)7-s + (−0.630 + 1.09i)9-s + (−0.652 + 0.376i)11-s + (−0.495 − 0.868i)13-s + (0.582 − 0.336i)15-s + (0.122 − 0.212i)17-s + (0.925 + 0.534i)19-s + 2.81i·21-s + (0.639 + 1.10i)23-s − 0.200·25-s + 0.391·27-s + (0.165 + 0.286i)29-s − 0.297i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 - 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.515 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2828601701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2828601701\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (1.78 + 3.13i)T \) |
| good | 3 | \( 1 + (1.30 + 2.25i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (4.29 + 2.48i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.16 - 1.25i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.505 + 0.874i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.03 - 2.32i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.06 - 5.31i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.890 - 1.54i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 1.65iT - 31T^{2} \) |
| 37 | \( 1 + (-5.54 + 3.20i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (10.4 - 6.01i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.67 + 2.89i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 6.68iT - 47T^{2} \) |
| 53 | \( 1 + 9.78T + 53T^{2} \) |
| 59 | \( 1 + (-10.6 - 6.16i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.68 + 9.83i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.37 + 1.94i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.68 + 3.28i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 9.35iT - 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 12.1iT - 83T^{2} \) |
| 89 | \( 1 + (-5.51 + 3.18i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.51 + 0.873i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.00141741945926441734656850288, −9.604824771762552283879596334409, −7.896831456369000578220520298242, −7.36944730042844927631394186981, −6.82212487453738095685625112055, −6.00151735713745246859882475082, −5.19375120600283733314104155606, −3.56463082610848414447384984275, −2.71884710021263934654434236974, −1.06807650056702961185879452280,
0.16359376819288072870186356529, 2.63887123795111661606124968293, 3.56361042060224145320269509427, 4.71164943481165068418711613939, 5.36691955068396964091313856088, 6.16974869157732189114652042688, 7.00662812378170643452074662847, 8.551467199646921179375814188511, 9.167418050071150025681701653805, 9.878932963003531149160775215883
