| 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
−11.82457521864337927962891874829, −10.67624396182314012979960670798, −9.902496124416297582628178239437, −9.012954043291767168870567896096, −8.074975266045120991402845997343, −6.80064425812161329406772793862, −5.62150895809157458602056373157, −4.33220603877290085023443093194, −2.74199822679308187102905812694, −1.96961119405330941606113766681,
1.62978132462123260359757832256, 3.68000681081568358044347141570, 4.87265355377144475076868289864, 5.73063836966683096992334723132, 7.17047032382926047450445914937, 7.977287434950321255159411896868, 8.953182236414017135102658379603, 9.719829824715017074132015995202, 10.67883195249226514150694011279, 12.00634840739321587316708379339
