| L(s) = 1 | + (1.30 − 0.541i)2-s + (0.804 − 4.04i)3-s + (1.41 − 1.41i)4-s + (−1.85 + 1.24i)5-s + (−1.13 − 5.71i)6-s + (−10.0 − 6.74i)7-s + (1.08 − 2.61i)8-s + (−7.39 − 3.06i)9-s + (−1.75 + 2.62i)10-s + (2.34 − 0.465i)11-s + (−4.58 − 6.85i)12-s + (9.02 + 9.02i)13-s + (−16.8 − 3.35i)14-s + (3.52 + 8.51i)15-s − 4i·16-s + (15.6 − 6.55i)17-s + ⋯ |
| L(s) = 1 | + (0.653 − 0.270i)2-s + (0.268 − 1.34i)3-s + (0.353 − 0.353i)4-s + (−0.371 + 0.248i)5-s + (−0.189 − 0.953i)6-s + (−1.44 − 0.963i)7-s + (0.135 − 0.326i)8-s + (−0.821 − 0.340i)9-s + (−0.175 + 0.262i)10-s + (0.212 − 0.0423i)11-s + (−0.381 − 0.571i)12-s + (0.693 + 0.693i)13-s + (−1.20 − 0.239i)14-s + (0.235 + 0.567i)15-s − 0.250i·16-s + (0.922 − 0.385i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.663 + 0.748i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.778337 - 1.73028i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.778337 - 1.73028i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.30 + 0.541i)T \) |
| 5 | \( 1 + (1.85 - 1.24i)T \) |
| 17 | \( 1 + (-15.6 + 6.55i)T \) |
| good | 3 | \( 1 + (-0.804 + 4.04i)T + (-8.31 - 3.44i)T^{2} \) |
| 7 | \( 1 + (10.0 + 6.74i)T + (18.7 + 45.2i)T^{2} \) |
| 11 | \( 1 + (-2.34 + 0.465i)T + (111. - 46.3i)T^{2} \) |
| 13 | \( 1 + (-9.02 - 9.02i)T + 169iT^{2} \) |
| 19 | \( 1 + (-14.9 + 6.17i)T + (255. - 255. i)T^{2} \) |
| 23 | \( 1 + (3.43 + 17.2i)T + (-488. + 202. i)T^{2} \) |
| 29 | \( 1 + (-17.2 - 25.7i)T + (-321. + 776. i)T^{2} \) |
| 31 | \( 1 + (47.8 + 9.51i)T + (887. + 367. i)T^{2} \) |
| 37 | \( 1 + (-5.56 + 27.9i)T + (-1.26e3 - 523. i)T^{2} \) |
| 41 | \( 1 + (-41.1 - 27.5i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-57.5 - 23.8i)T + (1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (34.9 + 34.9i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-25.8 + 10.7i)T + (1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (14.2 - 34.3i)T + (-2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (51.7 - 77.4i)T + (-1.42e3 - 3.43e3i)T^{2} \) |
| 67 | \( 1 - 115. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (4.12 - 20.7i)T + (-4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-24.4 + 16.3i)T + (2.03e3 - 4.92e3i)T^{2} \) |
| 79 | \( 1 + (-40.3 + 8.03i)T + (5.76e3 - 2.38e3i)T^{2} \) |
| 83 | \( 1 + (12.6 + 30.5i)T + (-4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (26.4 - 26.4i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (-19.2 - 28.8i)T + (-3.60e3 + 8.69e3i)T^{2} \) |
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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
−12.50363176994402744000000557489, −11.48671965756232258992238245843, −10.35753763929918625513722237236, −9.144967673642462605678210805884, −7.50513220896380321698481917529, −6.94778966593315602490164150844, −6.02106994056999044089918244601, −3.98142399047372832093650511461, −2.88584507526635030153104572518, −0.986724157443913076097743714727,
3.17402397993797938731030174292, 3.77786597973827347978521500656, 5.31360455290626603301362892193, 6.13054790633574051712676801675, 7.80642869210813216366163173018, 9.084324495969754353049922308484, 9.737690715844017974981312482924, 10.85980340999549397520819956334, 12.14999156994297578397739417729, 12.79444652367005745357021245948
