| L(s) = 1 | + (0.0747 + 0.997i)2-s + (0.955 + 0.294i)3-s + (−0.988 + 0.149i)4-s + (1.71 − 1.58i)5-s + (−0.222 + 0.974i)6-s + (1.81 − 1.92i)7-s + (−0.222 − 0.974i)8-s + (0.826 + 0.563i)9-s + (1.71 + 1.58i)10-s + (0.673 − 0.459i)11-s + (−0.988 − 0.149i)12-s + (−1.90 − 0.919i)13-s + (2.05 + 1.67i)14-s + (2.10 − 1.01i)15-s + (0.955 − 0.294i)16-s + (0.804 + 2.04i)17-s + ⋯ |
| L(s) = 1 | + (0.0528 + 0.705i)2-s + (0.551 + 0.170i)3-s + (−0.494 + 0.0745i)4-s + (0.764 − 0.709i)5-s + (−0.0908 + 0.398i)6-s + (0.687 − 0.726i)7-s + (−0.0786 − 0.344i)8-s + (0.275 + 0.187i)9-s + (0.540 + 0.501i)10-s + (0.203 − 0.138i)11-s + (−0.285 − 0.0430i)12-s + (−0.529 − 0.255i)13-s + (0.548 + 0.446i)14-s + (0.542 − 0.261i)15-s + (0.238 − 0.0736i)16-s + (0.195 + 0.497i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.68240 + 0.422980i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.68240 + 0.422980i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0747 - 0.997i)T \) |
| 3 | \( 1 + (-0.955 - 0.294i)T \) |
| 7 | \( 1 + (-1.81 + 1.92i)T \) |
| good | 5 | \( 1 + (-1.71 + 1.58i)T + (0.373 - 4.98i)T^{2} \) |
| 11 | \( 1 + (-0.673 + 0.459i)T + (4.01 - 10.2i)T^{2} \) |
| 13 | \( 1 + (1.90 + 0.919i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (-0.804 - 2.04i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (0.626 + 1.08i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.74 - 6.98i)T + (-16.8 - 15.6i)T^{2} \) |
| 29 | \( 1 + (3.99 - 5.01i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (-1.76 + 3.05i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (7.17 + 1.08i)T + (35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (-1.45 - 6.36i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-0.763 + 3.34i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (0.200 + 2.67i)T + (-46.4 + 7.00i)T^{2} \) |
| 53 | \( 1 + (-4.20 + 0.634i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (-7.28 - 6.75i)T + (4.40 + 58.8i)T^{2} \) |
| 61 | \( 1 + (12.6 + 1.90i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (-5.32 + 9.22i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.87 + 8.62i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (0.493 - 6.58i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (3.02 + 5.24i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.37 + 0.660i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (1.49 + 1.01i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
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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
−12.00634840739321587316708379339, −10.67883195249226514150694011279, −9.719829824715017074132015995202, −8.953182236414017135102658379603, −7.977287434950321255159411896868, −7.17047032382926047450445914937, −5.73063836966683096992334723132, −4.87265355377144475076868289864, −3.68000681081568358044347141570, −1.62978132462123260359757832256,
1.96961119405330941606113766681, 2.74199822679308187102905812694, 4.33220603877290085023443093194, 5.62150895809157458602056373157, 6.80064425812161329406772793862, 8.074975266045120991402845997343, 9.012954043291767168870567896096, 9.902496124416297582628178239437, 10.67624396182314012979960670798, 11.82457521864337927962891874829
