| L(s) = 1 | + (1.94 + 0.976i)2-s + (−1.52 + 0.815i)3-s + (1.63 + 2.19i)4-s + (−0.185 − 0.619i)5-s + (−3.76 + 0.0927i)6-s + (−0.724 − 1.67i)7-s + (0.277 + 1.57i)8-s + (1.67 − 2.49i)9-s + (0.244 − 1.38i)10-s + (−1.17 + 0.279i)11-s + (−4.28 − 2.02i)12-s + (4.45 + 2.93i)13-s + (0.231 − 3.97i)14-s + (0.788 + 0.795i)15-s + (0.571 − 1.90i)16-s + (−6.43 + 2.34i)17-s + ⋯ |
| L(s) = 1 | + (1.37 + 0.690i)2-s + (−0.882 + 0.470i)3-s + (0.816 + 1.09i)4-s + (−0.0829 − 0.276i)5-s + (−1.53 + 0.0378i)6-s + (−0.273 − 0.634i)7-s + (0.0980 + 0.556i)8-s + (0.556 − 0.830i)9-s + (0.0772 − 0.438i)10-s + (−0.354 + 0.0841i)11-s + (−1.23 − 0.583i)12-s + (1.23 + 0.812i)13-s + (0.0618 − 1.06i)14-s + (0.203 + 0.205i)15-s + (0.142 − 0.477i)16-s + (−1.56 + 0.567i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 - 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.548 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22009 + 0.659061i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22009 + 0.659061i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.52 - 0.815i)T \) |
| good | 2 | \( 1 + (-1.94 - 0.976i)T + (1.19 + 1.60i)T^{2} \) |
| 5 | \( 1 + (0.185 + 0.619i)T + (-4.17 + 2.74i)T^{2} \) |
| 7 | \( 1 + (0.724 + 1.67i)T + (-4.80 + 5.09i)T^{2} \) |
| 11 | \( 1 + (1.17 - 0.279i)T + (9.82 - 4.93i)T^{2} \) |
| 13 | \( 1 + (-4.45 - 2.93i)T + (5.14 + 11.9i)T^{2} \) |
| 17 | \( 1 + (6.43 - 2.34i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (5.97 + 2.17i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (1.23 - 2.86i)T + (-15.7 - 16.7i)T^{2} \) |
| 29 | \( 1 + (-0.342 - 5.87i)T + (-28.8 + 3.36i)T^{2} \) |
| 31 | \( 1 + (-2.75 + 0.321i)T + (30.1 - 7.14i)T^{2} \) |
| 37 | \( 1 + (1.09 + 0.918i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (-0.996 + 0.500i)T + (24.4 - 32.8i)T^{2} \) |
| 43 | \( 1 + (-6.61 - 7.00i)T + (-2.50 + 42.9i)T^{2} \) |
| 47 | \( 1 + (6.06 + 0.708i)T + (45.7 + 10.8i)T^{2} \) |
| 53 | \( 1 + (-4.26 + 7.38i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.03 - 0.481i)T + (52.7 + 26.4i)T^{2} \) |
| 61 | \( 1 + (2.14 - 2.87i)T + (-17.4 - 58.4i)T^{2} \) |
| 67 | \( 1 + (0.0709 - 1.21i)T + (-66.5 - 7.77i)T^{2} \) |
| 71 | \( 1 + (-1.41 + 8.02i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-1.11 - 6.32i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (12.7 + 6.39i)T + (47.1 + 63.3i)T^{2} \) |
| 83 | \( 1 + (-6.00 - 3.01i)T + (49.5 + 66.5i)T^{2} \) |
| 89 | \( 1 + (2.70 + 15.3i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-1.06 + 3.57i)T + (-81.0 - 53.3i)T^{2} \) |
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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
−14.60341634210838800971311797078, −13.30538452301054214014060688446, −12.81381509062848695997466650991, −11.44340988030109685559081257582, −10.51716657380255537045821332678, −8.827129730336586278947455264616, −6.84388956550885521958517427957, −6.20834820435939529890872204340, −4.70097604834625494349542747473, −3.93903585615150691712739038282,
2.48247281721473824599371340703, 4.32313581887129781453324422810, 5.70747510343222269539180622819, 6.51901384731959743394550732451, 8.447454973961182972508782124118, 10.59779336643050922115388790422, 11.12762796356034603248973068558, 12.25411980928819828528485211731, 13.00796679071340368030084216291, 13.69692314613554855124447635775
