| L(s) = 1 | + (0.726 − 0.495i)2-s + (−1.98 + 1.84i)3-s + (−0.448 + 1.14i)4-s + (1.60 + 0.496i)5-s + (−0.530 + 2.32i)6-s + (0.921 + 2.48i)7-s + (0.631 + 2.76i)8-s + (0.325 − 4.34i)9-s + (1.41 − 0.436i)10-s + (0.132 + 1.77i)11-s + (−1.21 − 3.09i)12-s + (0.900 + 0.433i)13-s + (1.89 + 1.34i)14-s + (−4.11 + 1.98i)15-s + (0.0288 + 0.0267i)16-s + (0.740 + 0.111i)17-s + ⋯ |
| L(s) = 1 | + (0.513 − 0.350i)2-s + (−1.14 + 1.06i)3-s + (−0.224 + 0.571i)4-s + (0.719 + 0.221i)5-s + (−0.216 + 0.949i)6-s + (0.348 + 0.937i)7-s + (0.223 + 0.977i)8-s + (0.108 − 1.44i)9-s + (0.447 − 0.137i)10-s + (0.0400 + 0.534i)11-s + (−0.351 − 0.894i)12-s + (0.249 + 0.120i)13-s + (0.507 + 0.359i)14-s + (−1.06 + 0.511i)15-s + (0.00721 + 0.00669i)16-s + (0.179 + 0.0270i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.464334 + 1.23241i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.464334 + 1.23241i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-0.921 - 2.48i)T \) |
| 13 | \( 1 + (-0.900 - 0.433i)T \) |
| good | 2 | \( 1 + (-0.726 + 0.495i)T + (0.730 - 1.86i)T^{2} \) |
| 3 | \( 1 + (1.98 - 1.84i)T + (0.224 - 2.99i)T^{2} \) |
| 5 | \( 1 + (-1.60 - 0.496i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (-0.132 - 1.77i)T + (-10.8 + 1.63i)T^{2} \) |
| 17 | \( 1 + (-0.740 - 0.111i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (-2.18 + 3.78i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.63 - 0.246i)T + (21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (3.15 - 3.95i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (-4.92 - 8.52i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.33 + 8.50i)T + (-27.1 + 25.1i)T^{2} \) |
| 41 | \( 1 + (1.25 + 5.50i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (1.31 - 5.76i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (-4.17 + 2.84i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (-0.0842 + 0.214i)T + (-38.8 - 36.0i)T^{2} \) |
| 59 | \( 1 + (-10.4 + 3.23i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (-4.18 - 10.6i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (2.33 + 4.04i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.535 - 0.671i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (10.8 + 7.38i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (0.397 - 0.688i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-12.6 + 6.07i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (0.103 - 1.37i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−10.98138130587508906630769846897, −10.26518893793307157661593844665, −9.315655879704952742290671124666, −8.623213607739092451537507669455, −7.20527431254510387483254278622, −5.96257009312420154507168881256, −5.25327363329070870151834531397, −4.62871610307494178153857377997, −3.48207970847299574203445540855, −2.18206537350070179116333191941,
0.73956569184413432364283453583, 1.67749249336147577229822595978, 3.91443754776071410564572818867, 5.09432443302985794483003896143, 5.84567805260345100575281174383, 6.32570879021723030805184572304, 7.29496394986119035698032989917, 8.168185492769781277913831574176, 9.720213134546339611393644148013, 10.25393226287561764416492780421
