| L(s) = 1 | + (−0.419 − 0.816i)2-s + (0.503 − 4.57i)3-s + (1.83 − 2.56i)4-s + (−5.01 + 2.34i)5-s + (−3.94 + 1.50i)6-s + (1.85 − 8.34i)7-s + (−6.49 − 0.956i)8-s + (−11.8 − 2.64i)9-s + (4.01 + 3.10i)10-s + (−5.44 − 3.05i)11-s + (−10.8 − 9.69i)12-s + (5.77 + 2.96i)13-s + (−7.59 + 1.98i)14-s + (8.22 + 24.0i)15-s + (−2.13 − 6.24i)16-s + (−9.26 + 16.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.209 − 0.408i)2-s + (0.167 − 1.52i)3-s + (0.459 − 0.642i)4-s + (−1.00 + 0.469i)5-s + (−0.657 + 0.251i)6-s + (0.265 − 1.19i)7-s + (−0.812 − 0.119i)8-s + (−1.32 − 0.294i)9-s + (0.401 + 0.310i)10-s + (−0.494 − 0.277i)11-s + (−0.901 − 0.807i)12-s + (0.444 + 0.228i)13-s + (−0.542 + 0.141i)14-s + (0.548 + 1.60i)15-s + (−0.133 − 0.390i)16-s + (−0.545 + 0.971i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 - 0.865i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.500 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.497299 + 0.861686i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.497299 + 0.861686i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 431 | \( 1 + (395. - 171. i)T \) |
| good | 2 | \( 1 + (0.419 + 0.816i)T + (-2.32 + 3.25i)T^{2} \) |
| 3 | \( 1 + (-0.503 + 4.57i)T + (-8.78 - 1.95i)T^{2} \) |
| 5 | \( 1 + (5.01 - 2.34i)T + (15.9 - 19.2i)T^{2} \) |
| 7 | \( 1 + (-1.85 + 8.34i)T + (-44.3 - 20.7i)T^{2} \) |
| 11 | \( 1 + (5.44 + 3.05i)T + (63.0 + 103. i)T^{2} \) |
| 13 | \( 1 + (-5.77 - 2.96i)T + (98.3 + 137. i)T^{2} \) |
| 17 | \( 1 + (9.26 - 16.5i)T + (-150. - 246. i)T^{2} \) |
| 19 | \( 1 + (-13.6 - 6.39i)T + (230. + 277. i)T^{2} \) |
| 23 | \( 1 + (-14.2 + 14.8i)T + (-19.3 - 528. i)T^{2} \) |
| 29 | \( 1 + (27.8 + 38.9i)T + (-271. + 795. i)T^{2} \) |
| 31 | \( 1 + (-5.59 + 14.6i)T + (-715. - 641. i)T^{2} \) |
| 37 | \( 1 + (-40.1 + 10.4i)T + (1.19e3 - 670. i)T^{2} \) |
| 41 | \( 1 + (-11.4 - 33.6i)T + (-1.33e3 + 1.02e3i)T^{2} \) |
| 43 | \( 1 + (-15.3 - 19.9i)T + (-467. + 1.78e3i)T^{2} \) |
| 47 | \( 1 + (-1.80 - 24.6i)T + (-2.18e3 + 321. i)T^{2} \) |
| 53 | \( 1 + (54.5 + 3.98i)T + (2.77e3 + 408. i)T^{2} \) |
| 59 | \( 1 + (23.8 + 46.3i)T + (-2.02e3 + 2.83e3i)T^{2} \) |
| 61 | \( 1 + (17.3 - 40.8i)T + (-2.58e3 - 2.67e3i)T^{2} \) |
| 67 | \( 1 + (11.5 - 3.02i)T + (3.91e3 - 2.19e3i)T^{2} \) |
| 71 | \( 1 + (4.96 + 67.7i)T + (-4.98e3 + 733. i)T^{2} \) |
| 73 | \( 1 + (-29.8 + 45.0i)T + (-2.08e3 - 4.90e3i)T^{2} \) |
| 79 | \( 1 + (-20.9 - 69.4i)T + (-5.20e3 + 3.44e3i)T^{2} \) |
| 83 | \( 1 + (37.5 + 4.13i)T + (6.72e3 + 1.49e3i)T^{2} \) |
| 89 | \( 1 + (-57.9 - 10.6i)T + (7.39e3 + 2.82e3i)T^{2} \) |
| 97 | \( 1 + (44.2 + 13.3i)T + (7.84e3 + 5.19e3i)T^{2} \) |
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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
−10.85039614011264403238531078829, −9.521420017147103690302323657144, −8.014710838496338964034065098696, −7.67901382870521426884651558746, −6.71757346241054304041074598313, −5.99660512521659565195538918586, −4.14286249312453269678193159210, −2.83210303467570236739941426198, −1.52139626717477113027983122977, −0.44245067079298002892072625836,
2.73857441789454484382047152202, 3.63249306277537770288436100763, 4.81858497440193107859139622322, 5.55985753653647814566614383373, 7.17041203278742969471126459191, 8.075914199964639838915809882631, 8.961095280024505664791628560485, 9.299360450113093874379410834869, 10.79457528164002130161054911614, 11.46630354294182938352027668414
