| L(s) = 1 | + (1.41 + 1.02i)2-s + (−1.57 + 3.54i)3-s + (−0.292 − 0.901i)4-s + (1.44 − 2.50i)5-s + (−5.87 + 3.39i)6-s + (2.88 − 3.20i)7-s + (2.67 − 8.22i)8-s + (−4.07 − 4.52i)9-s + (4.60 − 2.05i)10-s + (−2.25 + 10.6i)11-s + (3.65 + 0.384i)12-s + (−24.0 + 2.52i)13-s + (7.37 − 1.56i)14-s + (6.59 + 9.07i)15-s + (9.15 − 6.64i)16-s + (−3.12 − 14.7i)17-s + ⋯ |
| L(s) = 1 | + (0.706 + 0.513i)2-s + (−0.526 + 1.18i)3-s + (−0.0731 − 0.225i)4-s + (0.288 − 0.500i)5-s + (−0.979 + 0.565i)6-s + (0.412 − 0.458i)7-s + (0.333 − 1.02i)8-s + (−0.452 − 0.502i)9-s + (0.460 − 0.205i)10-s + (−0.205 + 0.965i)11-s + (0.304 + 0.0320i)12-s + (−1.84 + 0.194i)13-s + (0.526 − 0.111i)14-s + (0.439 + 0.604i)15-s + (0.572 − 0.415i)16-s + (−0.184 − 0.865i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.589 - 0.807i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.589 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.05740 + 0.536993i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05740 + 0.536993i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (-27.2 - 14.6i)T \) |
| good | 2 | \( 1 + (-1.41 - 1.02i)T + (1.23 + 3.80i)T^{2} \) |
| 3 | \( 1 + (1.57 - 3.54i)T + (-6.02 - 6.68i)T^{2} \) |
| 5 | \( 1 + (-1.44 + 2.50i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-2.88 + 3.20i)T + (-5.12 - 48.7i)T^{2} \) |
| 11 | \( 1 + (2.25 - 10.6i)T + (-110. - 49.2i)T^{2} \) |
| 13 | \( 1 + (24.0 - 2.52i)T + (165. - 35.1i)T^{2} \) |
| 17 | \( 1 + (3.12 + 14.7i)T + (-264. + 117. i)T^{2} \) |
| 19 | \( 1 + (0.850 - 8.09i)T + (-353. - 75.0i)T^{2} \) |
| 23 | \( 1 + (-22.7 - 7.38i)T + (427. + 310. i)T^{2} \) |
| 29 | \( 1 + (-28.5 + 39.3i)T + (-259. - 799. i)T^{2} \) |
| 37 | \( 1 + (29.4 - 16.9i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (22.6 - 10.0i)T + (1.12e3 - 1.24e3i)T^{2} \) |
| 43 | \( 1 + (29.0 + 3.04i)T + (1.80e3 + 384. i)T^{2} \) |
| 47 | \( 1 + (-5.63 + 4.09i)T + (682. - 2.10e3i)T^{2} \) |
| 53 | \( 1 + (-58.8 + 53.0i)T + (293. - 2.79e3i)T^{2} \) |
| 59 | \( 1 + (1.32 + 0.590i)T + (2.32e3 + 2.58e3i)T^{2} \) |
| 61 | \( 1 - 51.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (-10.0 + 17.3i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-32.7 - 36.4i)T + (-526. + 5.01e3i)T^{2} \) |
| 73 | \( 1 + (-14.7 + 69.4i)T + (-4.86e3 - 2.16e3i)T^{2} \) |
| 79 | \( 1 + (23.5 + 110. i)T + (-5.70e3 + 2.53e3i)T^{2} \) |
| 83 | \( 1 + (-23.7 - 53.4i)T + (-4.60e3 + 5.11e3i)T^{2} \) |
| 89 | \( 1 + (-66.7 + 21.6i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (53.1 + 163. i)T + (-7.61e3 + 5.53e3i)T^{2} \) |
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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
−16.71581077216193711802564379466, −15.49563781386551756187831902249, −14.76243813798799522780446201962, −13.48843618602451770960818710988, −12.00600516655483210322329870824, −10.25200910883828847831452561638, −9.606817573302218966666730146065, −7.13447433563227299905448902719, −5.05984647360360026544174419710, −4.66257990711246799676121462750,
2.57296741560342799342438942837, 5.19917590475408981237385952995, 6.87077764190521046307631783038, 8.358635964267173864637957758047, 10.71459502560347587051165663712, 11.96215921846376579855597717352, 12.64272128380042906163054808753, 13.77285688204350572859102319915, 14.86449318443549527489776645928, 16.98201182032849364504898834751
