| 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
−11.46630354294182938352027668414, −10.79457528164002130161054911614, −9.299360450113093874379410834869, −8.961095280024505664791628560485, −8.075914199964639838915809882631, −7.17041203278742969471126459191, −5.55985753653647814566614383373, −4.81858497440193107859139622322, −3.63249306277537770288436100763, −2.73857441789454484382047152202,
0.44245067079298002892072625836, 1.52139626717477113027983122977, 2.83210303467570236739941426198, 4.14286249312453269678193159210, 5.99660512521659565195538918586, 6.71757346241054304041074598313, 7.67901382870521426884651558746, 8.014710838496338964034065098696, 9.521420017147103690302323657144, 10.85039614011264403238531078829
