| L(s) = 1 | + (−3.68 + 3.68i)2-s + (−0.537 + 1.29i)3-s + 4.79i·4-s + (−80.0 − 33.1i)5-s + (−2.80 − 6.76i)6-s + (−92.4 + 38.2i)7-s + (−135. − 135. i)8-s + (170. + 170. i)9-s + (417. − 172. i)10-s + (122. + 296. i)11-s + (−6.22 − 2.57i)12-s + 440. i·13-s + (199. − 482. i)14-s + (86.0 − 86.0i)15-s + 847.·16-s + (26.8 − 1.19e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.651 + 0.651i)2-s + (−0.0344 + 0.0832i)3-s + 0.149i·4-s + (−1.43 − 0.593i)5-s + (−0.0317 − 0.0767i)6-s + (−0.713 + 0.295i)7-s + (−0.749 − 0.749i)8-s + (0.701 + 0.701i)9-s + (1.32 − 0.546i)10-s + (0.305 + 0.737i)11-s + (−0.0124 − 0.00516i)12-s + 0.722i·13-s + (0.272 − 0.657i)14-s + (0.0987 − 0.0987i)15-s + 0.827·16-s + (0.0225 − 0.999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0533i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.00969242 - 0.362868i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00969242 - 0.362868i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-26.8 + 1.19e3i)T \) |
| good | 2 | \( 1 + (3.68 - 3.68i)T - 32iT^{2} \) |
| 3 | \( 1 + (0.537 - 1.29i)T + (-171. - 171. i)T^{2} \) |
| 5 | \( 1 + (80.0 + 33.1i)T + (2.20e3 + 2.20e3i)T^{2} \) |
| 7 | \( 1 + (92.4 - 38.2i)T + (1.18e4 - 1.18e4i)T^{2} \) |
| 11 | \( 1 + (-122. - 296. i)T + (-1.13e5 + 1.13e5i)T^{2} \) |
| 13 | \( 1 - 440. iT - 3.71e5T^{2} \) |
| 19 | \( 1 + (1.69e3 - 1.69e3i)T - 2.47e6iT^{2} \) |
| 23 | \( 1 + (-144. - 348. i)T + (-4.55e6 + 4.55e6i)T^{2} \) |
| 29 | \( 1 + (1.81e3 + 751. i)T + (1.45e7 + 1.45e7i)T^{2} \) |
| 31 | \( 1 + (-1.35e3 + 3.26e3i)T + (-2.02e7 - 2.02e7i)T^{2} \) |
| 37 | \( 1 + (-547. + 1.32e3i)T + (-4.90e7 - 4.90e7i)T^{2} \) |
| 41 | \( 1 + (1.84e4 - 7.66e3i)T + (8.19e7 - 8.19e7i)T^{2} \) |
| 43 | \( 1 + (-9.78e3 - 9.78e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + 1.60e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + (1.70e4 - 1.70e4i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 + (-3.66e4 - 3.66e4i)T + 7.14e8iT^{2} \) |
| 61 | \( 1 + (-3.43e4 + 1.42e4i)T + (5.97e8 - 5.97e8i)T^{2} \) |
| 67 | \( 1 - 8.83e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-4.33e3 + 1.04e4i)T + (-1.27e9 - 1.27e9i)T^{2} \) |
| 73 | \( 1 + (3.04e4 + 1.26e4i)T + (1.46e9 + 1.46e9i)T^{2} \) |
| 79 | \( 1 + (-2.30e4 - 5.56e4i)T + (-2.17e9 + 2.17e9i)T^{2} \) |
| 83 | \( 1 + (2.40e4 - 2.40e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 - 2.11e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (9.21e4 + 3.81e4i)T + (6.07e9 + 6.07e9i)T^{2} \) |
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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
−18.60610431631964187948136323838, −16.76856759210903440212138616051, −16.15047290788411758682443402736, −15.20424607667293643007606069761, −12.84558281897630919764589319957, −11.84263991745236593464391670398, −9.610200619179912130178848603239, −8.179497228972832789772808956104, −7.00266363499866799605520978240, −4.15541664787006485780385035561,
0.35681518226734352105428224383, 3.52519217365411282996081164186, 6.67642854180739546690025223153, 8.550206732106301455380411621839, 10.28104759740474719117354114245, 11.31866494951517588575983366679, 12.69454423560431727866890659842, 14.84643083610288937730248691135, 15.72386366643349245996301452847, 17.53988310589923657122526453598
