| L(s) = 1 | + (−0.188 + 0.826i)2-s + (1.15 + 0.556i)4-s + (3.39 + 0.512i)5-s + (−0.134 + 0.232i)7-s + (−1.73 + 2.17i)8-s + (−1.06 + 2.71i)10-s + (2.96 − 1.42i)11-s + (−0.736 − 1.87i)13-s + (−0.167 − 0.155i)14-s + (0.129 + 0.161i)16-s + (−6.37 + 0.960i)17-s + (−0.449 − 5.99i)19-s + (3.63 + 2.48i)20-s + (0.621 + 2.72i)22-s + (1.83 + 1.25i)23-s + ⋯ |
| L(s) = 1 | + (−0.133 + 0.584i)2-s + (0.577 + 0.278i)4-s + (1.51 + 0.229i)5-s + (−0.0508 + 0.0880i)7-s + (−0.613 + 0.768i)8-s + (−0.336 + 0.857i)10-s + (0.894 − 0.430i)11-s + (−0.204 − 0.520i)13-s + (−0.0446 − 0.0414i)14-s + (0.0322 + 0.0404i)16-s + (−1.54 + 0.233i)17-s + (−0.103 − 1.37i)19-s + (0.813 + 0.554i)20-s + (0.132 + 0.580i)22-s + (0.382 + 0.260i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.54602 + 0.953091i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.54602 + 0.953091i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + (4.01 - 5.18i)T \) |
| good | 2 | \( 1 + (0.188 - 0.826i)T + (-1.80 - 0.867i)T^{2} \) |
| 5 | \( 1 + (-3.39 - 0.512i)T + (4.77 + 1.47i)T^{2} \) |
| 7 | \( 1 + (0.134 - 0.232i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.96 + 1.42i)T + (6.85 - 8.60i)T^{2} \) |
| 13 | \( 1 + (0.736 + 1.87i)T + (-9.52 + 8.84i)T^{2} \) |
| 17 | \( 1 + (6.37 - 0.960i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (0.449 + 5.99i)T + (-18.7 + 2.83i)T^{2} \) |
| 23 | \( 1 + (-1.83 - 1.25i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (1.75 + 1.63i)T + (2.16 + 28.9i)T^{2} \) |
| 31 | \( 1 + (4.93 - 1.52i)T + (25.6 - 17.4i)T^{2} \) |
| 37 | \( 1 + (-2.63 - 4.56i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.643 + 2.81i)T + (-36.9 - 17.7i)T^{2} \) |
| 47 | \( 1 + (-5.31 - 2.56i)T + (29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (2.34 - 5.97i)T + (-38.8 - 36.0i)T^{2} \) |
| 59 | \( 1 + (8.36 + 10.4i)T + (-13.1 + 57.5i)T^{2} \) |
| 61 | \( 1 + (9.50 + 2.93i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (0.950 + 12.6i)T + (-66.2 + 9.98i)T^{2} \) |
| 71 | \( 1 + (-4.98 + 3.40i)T + (25.9 - 66.0i)T^{2} \) |
| 73 | \( 1 + (-0.609 - 1.55i)T + (-53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 + (-6.97 + 12.0i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.60 + 5.19i)T + (6.20 - 82.7i)T^{2} \) |
| 89 | \( 1 + (-1.32 + 1.23i)T + (6.65 - 88.7i)T^{2} \) |
| 97 | \( 1 + (-2.36 + 1.13i)T + (60.4 - 75.8i)T^{2} \) |
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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
−11.20988817760994059445464888243, −10.75368676177642081363733535707, −9.289852633907289471289435256034, −8.976313119082747151751775651564, −7.58073514673091935376034774673, −6.46496594893154414548020846401, −6.17954500625584658022140434071, −4.90929523965339003121149585302, −3.03490671640133123521693945448, −1.96578618643163467952107209281,
1.59360219699502305273502143012, 2.33394327285996203991531294179, 4.02630090013616682294568782834, 5.50535232629114279627992769327, 6.39466756973797512712926048460, 7.09331444587848306709037744543, 8.917001649098590634927236442049, 9.454307093487183542479057916841, 10.24365310755585455159145868967, 11.04001934792127689984138364802
