| L(s) = 1 | + (−1.02 − 0.970i)2-s + (2.79 − 1.08i)3-s + (0.116 + 1.99i)4-s + (−0.238 + 2.03i)5-s + (−3.92 − 1.60i)6-s + (2.85 − 1.43i)7-s + (1.81 − 2.16i)8-s + (6.65 − 6.06i)9-s + (2.22 − 1.86i)10-s + (4.29 + 1.85i)11-s + (2.48 + 5.45i)12-s + (18.7 − 4.44i)13-s + (−4.33 − 1.29i)14-s + (1.54 + 5.95i)15-s + (−3.97 + 0.464i)16-s + (−16.9 + 2.98i)17-s + ⋯ |
| L(s) = 1 | + (−0.514 − 0.485i)2-s + (0.932 − 0.361i)3-s + (0.0290 + 0.499i)4-s + (−0.0476 + 0.407i)5-s + (−0.654 − 0.266i)6-s + (0.408 − 0.204i)7-s + (0.227 − 0.270i)8-s + (0.739 − 0.673i)9-s + (0.222 − 0.186i)10-s + (0.390 + 0.168i)11-s + (0.207 + 0.454i)12-s + (1.44 − 0.341i)13-s + (−0.309 − 0.0925i)14-s + (0.102 + 0.397i)15-s + (−0.248 + 0.0290i)16-s + (−0.997 + 0.175i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.689 + 0.724i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.689 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.51402 - 0.649320i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.51402 - 0.649320i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.02 + 0.970i)T \) |
| 3 | \( 1 + (-2.79 + 1.08i)T \) |
| good | 5 | \( 1 + (0.238 - 2.03i)T + (-24.3 - 5.76i)T^{2} \) |
| 7 | \( 1 + (-2.85 + 1.43i)T + (29.2 - 39.3i)T^{2} \) |
| 11 | \( 1 + (-4.29 - 1.85i)T + (83.0 + 88.0i)T^{2} \) |
| 13 | \( 1 + (-18.7 + 4.44i)T + (151. - 75.8i)T^{2} \) |
| 17 | \( 1 + (16.9 - 2.98i)T + (271. - 98.8i)T^{2} \) |
| 19 | \( 1 + (-0.592 + 3.35i)T + (-339. - 123. i)T^{2} \) |
| 23 | \( 1 + (-9.86 + 19.6i)T + (-315. - 424. i)T^{2} \) |
| 29 | \( 1 + (40.2 - 12.0i)T + (702. - 462. i)T^{2} \) |
| 31 | \( 1 + (-4.45 + 2.92i)T + (380. - 882. i)T^{2} \) |
| 37 | \( 1 + (27.1 + 9.87i)T + (1.04e3 + 879. i)T^{2} \) |
| 41 | \( 1 + (-1.32 + 1.25i)T + (97.7 - 1.67e3i)T^{2} \) |
| 43 | \( 1 + (-19.8 - 26.6i)T + (-530. + 1.77e3i)T^{2} \) |
| 47 | \( 1 + (28.5 - 43.4i)T + (-874. - 2.02e3i)T^{2} \) |
| 53 | \( 1 + (3.89 - 2.24i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (70.1 - 30.2i)T + (2.38e3 - 2.53e3i)T^{2} \) |
| 61 | \( 1 + (2.55 - 43.8i)T + (-3.69e3 - 431. i)T^{2} \) |
| 67 | \( 1 + (16.3 - 54.5i)T + (-3.75e3 - 2.46e3i)T^{2} \) |
| 71 | \( 1 + (-6.19 - 7.38i)T + (-875. + 4.96e3i)T^{2} \) |
| 73 | \( 1 + (44.9 + 37.6i)T + (925. + 5.24e3i)T^{2} \) |
| 79 | \( 1 + (49.2 - 52.1i)T + (-362. - 6.23e3i)T^{2} \) |
| 83 | \( 1 + (-64.2 - 60.6i)T + (400. + 6.87e3i)T^{2} \) |
| 89 | \( 1 + (-86.7 + 103. i)T + (-1.37e3 - 7.80e3i)T^{2} \) |
| 97 | \( 1 + (-90.7 + 10.6i)T + (9.15e3 - 2.16e3i)T^{2} \) |
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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
−12.65172567722744621021388219785, −11.24462250408504027514690458859, −10.61574976117853746449618421629, −9.199092476912814948093837146430, −8.573818331049451833218622141035, −7.47265284538064332915320374229, −6.44428259613178528490599196069, −4.22085507817183408131137514649, −2.99407375103733996414282939734, −1.45436526887939983301819966034,
1.69696910259462745504000105079, 3.70263517603484218008542180992, 5.05074154065342677442208017975, 6.55785497555715535012626524825, 7.83724428195440448141464487109, 8.811674724453905853276316487861, 9.212204638963536413943630085478, 10.61709937910804237110850204271, 11.51298586482584223398795343210, 13.12969128491589929333535644426
