| L(s) = 1 | + (−0.492 + 0.726i)2-s + (−0.0541 − 0.998i)3-s + (0.455 + 1.14i)4-s + (−3.82 − 1.77i)5-s + (0.751 + 0.452i)6-s + (−1.35 − 4.87i)7-s + (−2.76 − 0.609i)8-s + (−0.994 + 0.108i)9-s + (3.16 − 1.90i)10-s + (1.54 + 1.46i)11-s + (1.11 − 0.516i)12-s + (1.12 + 0.122i)13-s + (4.20 + 1.41i)14-s + (−1.56 + 3.91i)15-s + (0.0177 − 0.0168i)16-s + (0.0667 − 0.240i)17-s + ⋯ |
| L(s) = 1 | + (−0.348 + 0.513i)2-s + (−0.0312 − 0.576i)3-s + (0.227 + 0.571i)4-s + (−1.71 − 0.791i)5-s + (0.306 + 0.184i)6-s + (−0.511 − 1.84i)7-s + (−0.978 − 0.215i)8-s + (−0.331 + 0.0360i)9-s + (1.00 − 0.603i)10-s + (0.465 + 0.440i)11-s + (0.322 − 0.149i)12-s + (0.311 + 0.0338i)13-s + (1.12 + 0.378i)14-s + (−0.403 + 1.01i)15-s + (0.00443 − 0.00420i)16-s + (0.0161 − 0.0583i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.302 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.302 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.278926 - 0.381060i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.278926 - 0.381060i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.0541 + 0.998i)T \) |
| 59 | \( 1 + (-5.68 - 5.16i)T \) |
| good | 2 | \( 1 + (0.492 - 0.726i)T + (-0.740 - 1.85i)T^{2} \) |
| 5 | \( 1 + (3.82 + 1.77i)T + (3.23 + 3.81i)T^{2} \) |
| 7 | \( 1 + (1.35 + 4.87i)T + (-5.99 + 3.60i)T^{2} \) |
| 11 | \( 1 + (-1.54 - 1.46i)T + (0.595 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-1.12 - 0.122i)T + (12.6 + 2.79i)T^{2} \) |
| 17 | \( 1 + (-0.0667 + 0.240i)T + (-14.5 - 8.76i)T^{2} \) |
| 19 | \( 1 + (2.30 + 1.74i)T + (5.08 + 18.3i)T^{2} \) |
| 23 | \( 1 + (2.78 + 5.24i)T + (-12.9 + 19.0i)T^{2} \) |
| 29 | \( 1 + (-1.49 - 2.20i)T + (-10.7 + 26.9i)T^{2} \) |
| 31 | \( 1 + (-1.38 + 1.04i)T + (8.29 - 29.8i)T^{2} \) |
| 37 | \( 1 + (7.66 - 1.68i)T + (33.5 - 15.5i)T^{2} \) |
| 41 | \( 1 + (-2.32 + 4.37i)T + (-23.0 - 33.9i)T^{2} \) |
| 43 | \( 1 + (-1.98 + 1.88i)T + (2.32 - 42.9i)T^{2} \) |
| 47 | \( 1 + (-5.44 + 2.52i)T + (30.4 - 35.8i)T^{2} \) |
| 53 | \( 1 + (0.00505 + 0.00304i)T + (24.8 + 46.8i)T^{2} \) |
| 61 | \( 1 + (-7.77 + 11.4i)T + (-22.5 - 56.6i)T^{2} \) |
| 67 | \( 1 + (-3.91 - 0.862i)T + (60.8 + 28.1i)T^{2} \) |
| 71 | \( 1 + (-2.75 + 1.27i)T + (45.9 - 54.1i)T^{2} \) |
| 73 | \( 1 + (8.11 + 2.73i)T + (58.1 + 44.1i)T^{2} \) |
| 79 | \( 1 + (-0.206 + 3.79i)T + (-78.5 - 8.54i)T^{2} \) |
| 83 | \( 1 + (-0.687 + 4.19i)T + (-78.6 - 26.5i)T^{2} \) |
| 89 | \( 1 + (7.18 + 10.6i)T + (-32.9 + 82.6i)T^{2} \) |
| 97 | \( 1 + (-8.80 + 2.96i)T + (77.2 - 58.7i)T^{2} \) |
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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
−12.43223975701021393200310776929, −11.63186442342486911646476851806, −10.54831174338044480361019874496, −8.885665106659494295888477593390, −8.078922728441350171036074653086, −7.23179168674695849265173671603, −6.72330966145206901065594416916, −4.36199328753447531979676552667, −3.59267950986775332092441248006, −0.47555444108840252320170363182,
2.69589007951812594137863441975, 3.75944205064382694871005976618, 5.59918556815708887687329816468, 6.56721792149859110705627676494, 8.239315334789236648018717689316, 9.019894488305205841219996413584, 10.12097781168632759056911001238, 11.20423927014390970477915941970, 11.71499541550413126270866970126, 12.37105991990662194544310945812
