| L(s) = 1 | + (1.32 + 0.960i)2-s + (1.06 − 2.39i)3-s + (−0.411 − 1.26i)4-s + (−2.97 + 5.14i)5-s + (3.71 − 2.14i)6-s + (−7.48 + 8.30i)7-s + (2.69 − 8.28i)8-s + (1.40 + 1.56i)9-s + (−8.87 + 3.95i)10-s + (3.78 − 17.8i)11-s + (−3.47 − 0.365i)12-s + (3.33 − 0.350i)13-s + (−17.8 + 3.79i)14-s + (9.17 + 12.6i)15-s + (7.20 − 5.23i)16-s + (1.27 + 5.99i)17-s + ⋯ |
| L(s) = 1 | + (0.660 + 0.480i)2-s + (0.355 − 0.799i)3-s + (−0.102 − 0.316i)4-s + (−0.594 + 1.02i)5-s + (0.619 − 0.357i)6-s + (−1.06 + 1.18i)7-s + (0.336 − 1.03i)8-s + (0.156 + 0.173i)9-s + (−0.887 + 0.395i)10-s + (0.344 − 1.61i)11-s + (−0.289 − 0.0304i)12-s + (0.256 − 0.0269i)13-s + (−1.27 + 0.271i)14-s + (0.611 + 0.841i)15-s + (0.450 − 0.327i)16-s + (0.0749 + 0.352i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0961i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.26448 + 0.0609275i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26448 + 0.0609275i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (-7.17 - 30.1i)T \) |
| good | 2 | \( 1 + (-1.32 - 0.960i)T + (1.23 + 3.80i)T^{2} \) |
| 3 | \( 1 + (-1.06 + 2.39i)T + (-6.02 - 6.68i)T^{2} \) |
| 5 | \( 1 + (2.97 - 5.14i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (7.48 - 8.30i)T + (-5.12 - 48.7i)T^{2} \) |
| 11 | \( 1 + (-3.78 + 17.8i)T + (-110. - 49.2i)T^{2} \) |
| 13 | \( 1 + (-3.33 + 0.350i)T + (165. - 35.1i)T^{2} \) |
| 17 | \( 1 + (-1.27 - 5.99i)T + (-264. + 117. i)T^{2} \) |
| 19 | \( 1 + (-0.247 + 2.35i)T + (-353. - 75.0i)T^{2} \) |
| 23 | \( 1 + (14.7 + 4.78i)T + (427. + 310. i)T^{2} \) |
| 29 | \( 1 + (-5.18 + 7.13i)T + (-259. - 799. i)T^{2} \) |
| 37 | \( 1 + (54.9 - 31.7i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (21.1 - 9.40i)T + (1.12e3 - 1.24e3i)T^{2} \) |
| 43 | \( 1 + (-33.9 - 3.56i)T + (1.80e3 + 384. i)T^{2} \) |
| 47 | \( 1 + (-54.2 + 39.4i)T + (682. - 2.10e3i)T^{2} \) |
| 53 | \( 1 + (2.61 - 2.35i)T + (293. - 2.79e3i)T^{2} \) |
| 59 | \( 1 + (-8.47 - 3.77i)T + (2.32e3 + 2.58e3i)T^{2} \) |
| 61 | \( 1 + 24.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (29.0 - 50.2i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (60.9 + 67.6i)T + (-526. + 5.01e3i)T^{2} \) |
| 73 | \( 1 + (15.8 - 74.7i)T + (-4.86e3 - 2.16e3i)T^{2} \) |
| 79 | \( 1 + (15.1 + 71.2i)T + (-5.70e3 + 2.53e3i)T^{2} \) |
| 83 | \( 1 + (-8.38 - 18.8i)T + (-4.60e3 + 5.11e3i)T^{2} \) |
| 89 | \( 1 + (-37.6 + 12.2i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-27.1 - 83.5i)T + (-7.61e3 + 5.53e3i)T^{2} \) |
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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
−16.14563108478628389409676234629, −15.44451930994431347741177256456, −14.21097309112606738571882100250, −13.36349068877659033017203789467, −12.09289565840355154230768889465, −10.45489589126686448470136554905, −8.621703659523212050408964247255, −6.87248075511741648210492965294, −5.95282383164313121263367640081, −3.27003348077533580330055275922,
3.80007098983335320497438870111, 4.49203945480759627557315298520, 7.37508517203778388270169084528, 9.111457942670634684062242095023, 10.26398391310361268766379661914, 12.13617784369554732936113366981, 12.81991061228464255253033406978, 14.06620684530570244557194614902, 15.57768470842382411010081770151, 16.48563528317473403788828318473
