L(s) = 1 | + (−3.57 + 2.06i)2-s + (1.5 + 2.59i)3-s + (4.50 − 7.79i)4-s + 13.4i·5-s + (−10.7 − 6.18i)6-s + (−27.2 − 15.7i)7-s + 4.12i·8-s + (−4.5 + 7.79i)9-s + (−27.7 − 47.9i)10-s + (−35.0 + 20.2i)11-s + 27.0·12-s + (42.1 + 20.5i)13-s + 129.·14-s + (−34.9 + 20.1i)15-s + (27.4 + 47.6i)16-s + (−21.5 + 37.3i)17-s + ⋯ |
L(s) = 1 | + (−1.26 + 0.728i)2-s + (0.288 + 0.499i)3-s + (0.562 − 0.974i)4-s + 1.20i·5-s + (−0.728 − 0.420i)6-s + (−1.46 − 0.848i)7-s + 0.182i·8-s + (−0.166 + 0.288i)9-s + (−0.876 − 1.51i)10-s + (−0.961 + 0.555i)11-s + 0.649·12-s + (0.898 + 0.438i)13-s + 2.47·14-s + (−0.601 + 0.347i)15-s + (0.429 + 0.744i)16-s + (−0.307 + 0.533i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0573i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0128761 - 0.448686i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0128761 - 0.448686i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.5 - 2.59i)T \) |
| 13 | \( 1 + (-42.1 - 20.5i)T \) |
good | 2 | \( 1 + (3.57 - 2.06i)T + (4 - 6.92i)T^{2} \) |
| 5 | \( 1 - 13.4iT - 125T^{2} \) |
| 7 | \( 1 + (27.2 + 15.7i)T + (171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (35.0 - 20.2i)T + (665.5 - 1.15e3i)T^{2} \) |
| 17 | \( 1 + (21.5 - 37.3i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-23.3 - 13.4i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (9.50 + 16.4i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-77.0 - 133. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 308. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (37.6 - 21.7i)T + (2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-41.4 + 23.9i)T + (3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-171. + 296. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + 133. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 438.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-511. - 295. i)T + (1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-270. + 468. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (199. - 115. i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (389. + 224. i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 - 389. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 897.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.30e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (-801. + 462. i)T + (3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-1.35e3 - 780. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−16.09003191765808513741895079112, −15.77122176988081621379118791478, −14.35195807109856558262729279908, −13.01016566552357720410455458100, −10.50422709266487798097014986255, −10.27811045048479165909652027449, −8.866562938068789930750052983620, −7.29763802444551228660047312993, −6.47291460446494201590771646096, −3.44301783024545184853557167260,
0.53377136543492187555968545806, 2.75537342279292489765590682191, 5.82520214043378509770827240192, 8.020645559262573432685455299044, 8.947753212563347967369678117224, 9.791034634738723748287434986914, 11.43854107125190542435127332301, 12.68859128094707538602600611360, 13.37555518851625846592237256598, 15.71790626734812680290644657218