| L(s) = 1 | + (−0.352 + 0.542i)2-s + (0.913 + 0.406i)3-s + (0.642 + 1.44i)4-s + (−0.201 + 0.396i)5-s + (−0.542 + 0.352i)6-s + (−4.51 − 1.73i)7-s + (−2.28 − 0.362i)8-s + (0.669 + 0.743i)9-s + (−0.143 − 0.249i)10-s + (−2.02 + 2.62i)11-s + 1.58i·12-s + (−0.113 + 3.60i)13-s + (2.53 − 1.84i)14-s + (−0.345 + 0.279i)15-s + (−1.11 + 1.23i)16-s + (−5.66 − 1.20i)17-s + ⋯ |
| L(s) = 1 | + (−0.249 + 0.383i)2-s + (0.527 + 0.234i)3-s + (0.321 + 0.722i)4-s + (−0.0902 + 0.177i)5-s + (−0.221 + 0.143i)6-s + (−1.70 − 0.655i)7-s + (−0.809 − 0.128i)8-s + (0.223 + 0.247i)9-s + (−0.0455 − 0.0788i)10-s + (−0.611 + 0.790i)11-s + 0.456i·12-s + (−0.0316 + 0.999i)13-s + (0.677 − 0.492i)14-s + (−0.0892 + 0.0722i)15-s + (−0.277 + 0.308i)16-s + (−1.37 − 0.291i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.351i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 - 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.157296 + 0.866231i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.157296 + 0.866231i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.913 - 0.406i)T \) |
| 11 | \( 1 + (2.02 - 2.62i)T \) |
| 13 | \( 1 + (0.113 - 3.60i)T \) |
| good | 2 | \( 1 + (0.352 - 0.542i)T + (-0.813 - 1.82i)T^{2} \) |
| 5 | \( 1 + (0.201 - 0.396i)T + (-2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (4.51 + 1.73i)T + (5.20 + 4.68i)T^{2} \) |
| 17 | \( 1 + (5.66 + 1.20i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-3.63 - 2.94i)T + (3.95 + 18.5i)T^{2} \) |
| 23 | \( 1 + (0.797 - 0.460i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.73 - 0.813i)T + (28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (-8.75 + 4.45i)T + (18.2 - 25.0i)T^{2} \) |
| 37 | \( 1 + (1.12 + 1.39i)T + (-7.69 + 36.1i)T^{2} \) |
| 41 | \( 1 + (10.2 - 3.91i)T + (30.4 - 27.4i)T^{2} \) |
| 43 | \( 1 + (-1.10 + 1.91i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.142 - 0.896i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (1.28 - 3.95i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-1.47 + 3.84i)T + (-43.8 - 39.4i)T^{2} \) |
| 61 | \( 1 + (0.979 - 4.60i)T + (-55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (-0.604 + 2.25i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (4.11 + 6.32i)T + (-28.8 + 64.8i)T^{2} \) |
| 73 | \( 1 + (-0.473 - 2.98i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-8.22 - 2.67i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-4.63 - 2.36i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (6.95 + 1.86i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (0.698 - 13.3i)T + (-96.4 - 10.1i)T^{2} \) |
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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
−11.65166397630192405971239272953, −10.37307659839996801804282167506, −9.653772034971141689132091334275, −8.901013980237919742287471070094, −7.77496051684141456390586350720, −6.90158271687292315594707200333, −6.46219505669588827855744944283, −4.53312539669013252690352272056, −3.47015451642301635333299403754, −2.56733789042950871781122498551,
0.53051095875033068410179377841, 2.62994314045501085114455046506, 3.10803124177560321825105854043, 5.06925528660978593794590227155, 6.24652285983438876501141825339, 6.75039360788799569734111279806, 8.408553153540006292172222637980, 8.927441979958083154335542172460, 10.03758697401219007323582529491, 10.42986128006147894336299591717
