| L(s) = 1 | + (−3.23 + 1.33i)2-s + (−0.783 + 3.93i)3-s + (5.81 − 5.81i)4-s + (0.268 − 0.179i)5-s + (−2.73 − 13.7i)6-s + (5.55 + 3.71i)7-s + (−5.65 + 13.6i)8-s + (−6.57 − 2.72i)9-s + (−0.627 + 0.939i)10-s + (5.27 − 1.04i)11-s + (18.3 + 27.4i)12-s + (−12.1 − 12.1i)13-s + (−22.9 − 4.56i)14-s + (0.496 + 1.19i)15-s − 18.7i·16-s + (15.7 − 6.50i)17-s + ⋯ |
| L(s) = 1 | + (−1.61 + 0.669i)2-s + (−0.261 + 1.31i)3-s + (1.45 − 1.45i)4-s + (0.0537 − 0.0359i)5-s + (−0.456 − 2.29i)6-s + (0.794 + 0.530i)7-s + (−0.706 + 1.70i)8-s + (−0.730 − 0.302i)9-s + (−0.0627 + 0.0939i)10-s + (0.479 − 0.0953i)11-s + (1.52 + 2.28i)12-s + (−0.934 − 0.934i)13-s + (−1.63 − 0.325i)14-s + (0.0331 + 0.0799i)15-s − 1.17i·16-s + (0.923 − 0.382i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.250 - 0.968i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.250 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.261096 + 0.337182i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.261096 + 0.337182i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-15.7 + 6.50i)T \) |
| good | 2 | \( 1 + (3.23 - 1.33i)T + (2.82 - 2.82i)T^{2} \) |
| 3 | \( 1 + (0.783 - 3.93i)T + (-8.31 - 3.44i)T^{2} \) |
| 5 | \( 1 + (-0.268 + 0.179i)T + (9.56 - 23.0i)T^{2} \) |
| 7 | \( 1 + (-5.55 - 3.71i)T + (18.7 + 45.2i)T^{2} \) |
| 11 | \( 1 + (-5.27 + 1.04i)T + (111. - 46.3i)T^{2} \) |
| 13 | \( 1 + (12.1 + 12.1i)T + 169iT^{2} \) |
| 19 | \( 1 + (-9.69 + 4.01i)T + (255. - 255. i)T^{2} \) |
| 23 | \( 1 + (1.90 + 9.56i)T + (-488. + 202. i)T^{2} \) |
| 29 | \( 1 + (21.5 + 32.2i)T + (-321. + 776. i)T^{2} \) |
| 31 | \( 1 + (-17.4 - 3.47i)T + (887. + 367. i)T^{2} \) |
| 37 | \( 1 + (-3.17 + 15.9i)T + (-1.26e3 - 523. i)T^{2} \) |
| 41 | \( 1 + (-11.1 - 7.43i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (57.3 + 23.7i)T + (1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-23.9 - 23.9i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (17.1 - 7.11i)T + (1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (33.8 - 81.7i)T + (-2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-36.8 + 55.1i)T + (-1.42e3 - 3.43e3i)T^{2} \) |
| 67 | \( 1 - 24.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-2.55 + 12.8i)T + (-4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-64.3 + 43.0i)T + (2.03e3 - 4.92e3i)T^{2} \) |
| 79 | \( 1 + (94.8 - 18.8i)T + (5.76e3 - 2.38e3i)T^{2} \) |
| 83 | \( 1 + (11.4 + 27.6i)T + (-4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-12.6 + 12.6i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (-19.6 - 29.3i)T + (-3.60e3 + 8.69e3i)T^{2} \) |
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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
−18.78864196516266003171917231546, −17.51308853649557702801537810229, −16.73497563853248727845058356743, −15.52572184968351185055295866148, −14.77731447320012769023750783438, −11.54682955690271112269840902675, −10.20038787692001061512764155446, −9.285770438807415051262676380522, −7.75971450553694670606486963475, −5.41676655484049095376765873050,
1.57473095095843119857374504030, 7.03375733370396620176314312845, 8.050495860723100125927459061493, 9.826565924060754714830581117875, 11.45458084085890860464998728999, 12.29481679610616864042980698138, 14.18957761202667083016975392068, 16.69806869613767616091702009159, 17.48116627125057579123384309554, 18.41937942185387413293903838138
