| L(s) = 1 | + (−1.09 − 0.889i)2-s + (−0.323 − 0.133i)3-s + (0.417 + 1.95i)4-s + (−0.705 + 0.359i)5-s + (0.236 + 0.434i)6-s + (−2.28 − 3.73i)7-s + (1.28 − 2.52i)8-s + (−2.03 − 2.03i)9-s + (1.09 + 0.232i)10-s + (2.94 + 0.231i)11-s + (0.127 − 0.688i)12-s + (−1.05 − 4.39i)13-s + (−0.806 + 6.14i)14-s + (0.276 − 0.0217i)15-s + (−3.65 + 1.63i)16-s + (−2.76 + 2.36i)17-s + ⋯ |
| L(s) = 1 | + (−0.777 − 0.629i)2-s + (−0.186 − 0.0773i)3-s + (0.208 + 0.977i)4-s + (−0.315 + 0.160i)5-s + (0.0964 + 0.177i)6-s + (−0.865 − 1.41i)7-s + (0.453 − 0.891i)8-s + (−0.678 − 0.678i)9-s + (0.346 + 0.0735i)10-s + (0.886 + 0.0697i)11-s + (0.0366 − 0.198i)12-s + (−0.292 − 1.21i)13-s + (−0.215 + 1.64i)14-s + (0.0713 − 0.00561i)15-s + (−0.912 + 0.408i)16-s + (−0.671 + 0.573i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 + 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.782 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.159797 - 0.456929i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.159797 - 0.456929i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.09 + 0.889i)T \) |
| 41 | \( 1 + (-1.56 + 6.20i)T \) |
| good | 3 | \( 1 + (0.323 + 0.133i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (0.705 - 0.359i)T + (2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (2.28 + 3.73i)T + (-3.17 + 6.23i)T^{2} \) |
| 11 | \( 1 + (-2.94 - 0.231i)T + (10.8 + 1.72i)T^{2} \) |
| 13 | \( 1 + (1.05 + 4.39i)T + (-11.5 + 5.90i)T^{2} \) |
| 17 | \( 1 + (2.76 - 2.36i)T + (2.65 - 16.7i)T^{2} \) |
| 19 | \( 1 + (4.50 + 1.08i)T + (16.9 + 8.62i)T^{2} \) |
| 23 | \( 1 + (-6.50 + 4.72i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-5.31 - 4.54i)T + (4.53 + 28.6i)T^{2} \) |
| 31 | \( 1 + (0.131 + 0.403i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.21 + 6.82i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (-3.65 - 0.579i)T + (40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (4.08 - 6.66i)T + (-21.3 - 41.8i)T^{2} \) |
| 53 | \( 1 + (1.61 - 1.88i)T + (-8.29 - 52.3i)T^{2} \) |
| 59 | \( 1 + (2.29 + 3.16i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-12.3 + 1.96i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (0.516 + 6.55i)T + (-66.1 + 10.4i)T^{2} \) |
| 71 | \( 1 + (0.111 - 1.41i)T + (-70.1 - 11.1i)T^{2} \) |
| 73 | \( 1 + (-2.08 + 2.08i)T - 73iT^{2} \) |
| 79 | \( 1 + (3.24 - 7.83i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + 5.01iT - 83T^{2} \) |
| 89 | \( 1 + (5.31 - 3.25i)T + (40.4 - 79.2i)T^{2} \) |
| 97 | \( 1 + (-0.157 - 1.99i)T + (-95.8 + 15.1i)T^{2} \) |
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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
−12.54604416667532828606424567486, −11.06750607725032520614049863267, −10.66072714954291028734125783404, −9.481317444934882646297933649460, −8.522191043622825260616168148581, −7.19142035839037023328724336498, −6.46200882093521260191815172370, −4.12880106971535956073368497451, −3.10281863537772893192254567634, −0.58853650640669355821005295420,
2.39924208664876617132766357701, 4.73011520816530884110924205803, 6.04140065046269696414344920930, 6.78654607995597352729572124565, 8.373341805910280821760129271880, 9.022761760686954208928325836565, 9.859540864136445880984272167981, 11.41794820988412822007402484605, 11.81598165489996639613196516877, 13.34575605585833176411482098188
