| L(s) = 1 | + (2.87 + 0.507i)2-s + (−1.11 + 1.32i)3-s + (4.27 + 1.55i)4-s + (1.04 − 0.379i)5-s + (−3.87 + 3.25i)6-s + (−1.13 − 1.96i)7-s + (1.37 + 0.796i)8-s + (−0.520 − 2.95i)9-s + (3.19 − 0.562i)10-s + (−0.705 + 1.22i)11-s + (−6.81 + 3.93i)12-s + (−6.19 − 7.38i)13-s + (−2.27 − 6.24i)14-s + (−0.656 + 1.80i)15-s + (−10.3 − 8.69i)16-s + (0.749 − 4.24i)17-s + ⋯ |
| L(s) = 1 | + (1.43 + 0.253i)2-s + (−0.371 + 0.442i)3-s + (1.06 + 0.388i)4-s + (0.208 − 0.0758i)5-s + (−0.646 + 0.542i)6-s + (−0.162 − 0.281i)7-s + (0.172 + 0.0995i)8-s + (−0.0578 − 0.328i)9-s + (0.319 − 0.0562i)10-s + (−0.0640 + 0.111i)11-s + (−0.568 + 0.327i)12-s + (−0.476 − 0.568i)13-s + (−0.162 − 0.446i)14-s + (−0.0437 + 0.120i)15-s + (−0.647 − 0.543i)16-s + (0.0440 − 0.249i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.491i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.870 - 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.93307 + 0.508313i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.93307 + 0.508313i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.11 - 1.32i)T \) |
| 19 | \( 1 + (-0.262 - 18.9i)T \) |
| good | 2 | \( 1 + (-2.87 - 0.507i)T + (3.75 + 1.36i)T^{2} \) |
| 5 | \( 1 + (-1.04 + 0.379i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (1.13 + 1.96i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (0.705 - 1.22i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (6.19 + 7.38i)T + (-29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-0.749 + 4.24i)T + (-271. - 98.8i)T^{2} \) |
| 23 | \( 1 + (-34.8 - 12.6i)T + (405. + 340. i)T^{2} \) |
| 29 | \( 1 + (-41.2 + 7.26i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (44.1 - 25.5i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 1.97iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-0.341 + 0.406i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-39.8 + 14.4i)T + (1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-11.2 - 63.9i)T + (-2.07e3 + 755. i)T^{2} \) |
| 53 | \( 1 + (-14.3 + 39.4i)T + (-2.15e3 - 1.80e3i)T^{2} \) |
| 59 | \( 1 + (43.9 + 7.74i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (50.0 + 18.2i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (21.2 - 3.75i)T + (4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (37.3 + 102. i)T + (-3.86e3 + 3.24e3i)T^{2} \) |
| 73 | \( 1 + (42.0 + 35.2i)T + (925. + 5.24e3i)T^{2} \) |
| 79 | \( 1 + (-20.1 + 23.9i)T + (-1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-47.5 - 82.4i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (34.4 + 41.1i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (73.8 + 13.0i)T + (8.84e3 + 3.21e3i)T^{2} \) |
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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
−14.95385815799145562289578961337, −13.99071703487365805207629373624, −12.91339786566876925564044887610, −12.03804679453553399580972932334, −10.69298260336675807708207543914, −9.361739028044036082139934944053, −7.29646078511109222865967958369, −5.86408706101924271291876153035, −4.84347842632476287824144231374, −3.35462694956772237929627291737,
2.61479044351044351592685839885, 4.55041434432989967847039067545, 5.82686672078873131094575862247, 7.00161867216013166260108704360, 8.988431213753450229455810713945, 10.79097848935703028148835246622, 11.85035725497904436131155946943, 12.72888898959323176607904700270, 13.60658103710138965941478999383, 14.58763695740413293207899499996
