| L(s) = 1 | + (0.519 + 0.334i)2-s + (0.318 + 2.21i)3-s + (−0.672 − 1.47i)4-s + (−1.18 − 0.348i)5-s + (−0.574 + 1.25i)6-s + (−2.11 + 2.44i)7-s + (0.318 − 2.21i)8-s + (−1.91 + 0.563i)9-s + (−0.500 − 0.577i)10-s + (−4.40 + 2.83i)11-s + (3.04 − 1.95i)12-s + (−1.96 − 2.26i)13-s + (−1.91 + 0.563i)14-s + (0.393 − 2.73i)15-s + (−1.21 + 1.40i)16-s + (0.317 − 0.694i)17-s + ⋯ |
| L(s) = 1 | + (0.367 + 0.236i)2-s + (0.183 + 1.27i)3-s + (−0.336 − 0.735i)4-s + (−0.530 − 0.155i)5-s + (−0.234 + 0.513i)6-s + (−0.800 + 0.924i)7-s + (0.112 − 0.782i)8-s + (−0.639 + 0.187i)9-s + (−0.158 − 0.182i)10-s + (−1.32 + 0.853i)11-s + (0.878 − 0.564i)12-s + (−0.544 − 0.628i)13-s + (−0.512 + 0.150i)14-s + (0.101 − 0.706i)15-s + (−0.303 + 0.350i)16-s + (0.0769 − 0.168i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.945 + 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0712092 - 0.425362i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0712092 - 0.425362i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 + (-0.519 - 0.334i)T + (0.830 + 1.81i)T^{2} \) |
| 3 | \( 1 + (-0.318 - 2.21i)T + (-2.87 + 0.845i)T^{2} \) |
| 5 | \( 1 + (1.18 + 0.348i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (2.11 - 2.44i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (4.40 - 2.83i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (1.96 + 2.26i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.317 + 0.694i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (0.830 + 1.81i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (1.24 - 2.72i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.954 - 6.63i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-1.18 + 0.348i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-3.33 - 0.978i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 2.23T + 47T^{2} \) |
| 53 | \( 1 + (0.309 - 0.356i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (4.23 + 4.89i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.988 + 6.87i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (2.32 + 1.49i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-10.2 - 6.61i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-2.71 - 5.93i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-7.16 - 8.27i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-8.40 + 2.46i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.49 - 10.3i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (16.9 + 4.98i)T + (81.6 + 52.4i)T^{2} \) |
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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
−11.02022660847783524750747445975, −10.22244663905117805680195815704, −9.688164833962705431702906243002, −8.999249304339084388962873935044, −7.83599418222507806080056485351, −6.59717871704066875809974348450, −5.26873176264432105993142123032, −4.99465427944125787372890134012, −3.83109706711709432344431835825, −2.65006932074301751933721650524,
0.20590961816314031028362261275, 2.31453573086919654181379519262, 3.37626308693224786713652983591, 4.32822978940052604936863683627, 5.82457940513799956578682825964, 6.97457501271528735664310093958, 7.75668813238176135680668173414, 8.060789496312296649183144368254, 9.406159278111202041287104085147, 10.56720605707525157818480767379
