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
−10.25393226287561764416492780421, −9.720213134546339611393644148013, −8.168185492769781277913831574176, −7.29496394986119035698032989917, −6.32570879021723030805184572304, −5.84567805260345100575281174383, −5.09432443302985794483003896143, −3.91443754776071410564572818867, −1.67749249336147577229822595978, −0.73956569184413432364283453583,
2.18206537350070179116333191941, 3.48207970847299574203445540855, 4.62871610307494178153857377997, 5.25327363329070870151834531397, 5.96257009312420154507168881256, 7.20527431254510387483254278622, 8.623213607739092451537507669455, 9.315655879704952742290671124666, 10.26518893793307157661593844665, 10.98138130587508906630769846897