| L(s) = 1 | + (−0.0988 − 0.484i)2-s + (0.987 + 0.160i)3-s + (1.61 − 0.688i)4-s + (0.576 − 0.142i)5-s + (−0.0199 − 0.494i)6-s + (1.32 − 2.79i)7-s + (−1.05 − 1.52i)8-s + (0.948 + 0.316i)9-s + (−0.125 − 0.265i)10-s + (−0.422 + 0.141i)11-s + (1.70 − 0.420i)12-s + (−2.93 − 2.09i)13-s + (−1.48 − 0.366i)14-s + (0.592 − 0.0478i)15-s + (1.79 − 1.87i)16-s + (−1.94 + 4.09i)17-s + ⋯ |
| L(s) = 1 | + (−0.0699 − 0.342i)2-s + (0.569 + 0.0926i)3-s + (0.807 − 0.344i)4-s + (0.257 − 0.0635i)5-s + (−0.00812 − 0.201i)6-s + (0.501 − 1.05i)7-s + (−0.372 − 0.540i)8-s + (0.316 + 0.105i)9-s + (−0.0398 − 0.0839i)10-s + (−0.127 + 0.0425i)11-s + (0.492 − 0.121i)12-s + (−0.813 − 0.581i)13-s + (−0.397 − 0.0978i)14-s + (0.152 − 0.0123i)15-s + (0.449 − 0.467i)16-s + (−0.471 + 0.994i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.75474 - 1.09611i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.75474 - 1.09611i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.987 - 0.160i)T \) |
| 13 | \( 1 + (2.93 + 2.09i)T \) |
| good | 2 | \( 1 + (0.0988 + 0.484i)T + (-1.83 + 0.783i)T^{2} \) |
| 5 | \( 1 + (-0.576 + 0.142i)T + (4.42 - 2.32i)T^{2} \) |
| 7 | \( 1 + (-1.32 + 2.79i)T + (-4.42 - 5.42i)T^{2} \) |
| 11 | \( 1 + (0.422 - 0.141i)T + (8.79 - 6.60i)T^{2} \) |
| 17 | \( 1 + (1.94 - 4.09i)T + (-10.7 - 13.1i)T^{2} \) |
| 19 | \( 1 + (-0.0631 - 0.109i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.186 + 0.323i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.64 - 8.06i)T + (-26.6 + 11.3i)T^{2} \) |
| 31 | \( 1 + (-3.99 - 2.09i)T + (17.6 + 25.5i)T^{2} \) |
| 37 | \( 1 + (-2.94 + 1.86i)T + (15.8 - 33.4i)T^{2} \) |
| 41 | \( 1 + (-6.77 - 1.10i)T + (38.8 + 12.9i)T^{2} \) |
| 43 | \( 1 + (-0.513 - 0.324i)T + (18.4 + 38.8i)T^{2} \) |
| 47 | \( 1 + (-0.123 - 1.01i)T + (-45.6 + 11.2i)T^{2} \) |
| 53 | \( 1 + (1.97 + 2.86i)T + (-18.7 + 49.5i)T^{2} \) |
| 59 | \( 1 + (-0.928 - 0.966i)T + (-2.37 + 58.9i)T^{2} \) |
| 61 | \( 1 + (0.641 + 0.0517i)T + (60.2 + 9.78i)T^{2} \) |
| 67 | \( 1 + (-7.11 - 3.03i)T + (46.4 + 48.3i)T^{2} \) |
| 71 | \( 1 + (2.58 - 3.16i)T + (-14.2 - 69.5i)T^{2} \) |
| 73 | \( 1 + (5.21 + 4.62i)T + (8.79 + 72.4i)T^{2} \) |
| 79 | \( 1 + (1.10 + 9.08i)T + (-76.7 + 18.9i)T^{2} \) |
| 83 | \( 1 + (-2.06 + 5.43i)T + (-62.1 - 55.0i)T^{2} \) |
| 89 | \( 1 + (2.85 - 4.93i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.25 - 7.77i)T + (-81.9 + 51.8i)T^{2} \) |
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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
−10.54037951459461524037668468779, −10.18176694592743730846793948948, −9.122553167040883307082314014079, −7.904284354451323272535379013140, −7.27488355704391920837493007502, −6.23045838757128831284341222103, −4.97306974811253435972544992844, −3.75989828224661716308543431278, −2.53985825389696247517035796677, −1.33243857351570488115217126989,
2.19111116569187065353699614422, 2.67553662132152456602292146699, 4.40480505972086287663753000875, 5.65149192931960426246920467155, 6.56396654242004396509951786571, 7.56250136326935212812226333896, 8.239758332055742384258529818634, 9.186412462713390303412141880996, 9.975015625446789051577067118804, 11.38574997659673207705828194842
