L(s) = 1 | + (1.54 − 1.26i)2-s + (−0.126 + 0.219i)3-s + (0.785 − 3.92i)4-s + (1.78 + 3.09i)5-s + (0.0821 + 0.499i)6-s + (−3.75 − 7.06i)8-s + (4.46 + 7.73i)9-s + (6.68 + 2.52i)10-s + (6.82 + 3.94i)11-s + (0.760 + 0.668i)12-s + 18.1·13-s − 0.904·15-s + (−14.7 − 6.16i)16-s + (8.26 + 4.76i)17-s + (16.7 + 6.30i)18-s + (−12.4 − 21.4i)19-s + ⋯ |
L(s) = 1 | + (0.773 − 0.633i)2-s + (−0.0422 + 0.0731i)3-s + (0.196 − 0.980i)4-s + (0.357 + 0.618i)5-s + (0.0136 + 0.0833i)6-s + (−0.469 − 0.882i)8-s + (0.496 + 0.859i)9-s + (0.668 + 0.252i)10-s + (0.620 + 0.358i)11-s + (0.0633 + 0.0557i)12-s + 1.39·13-s − 0.0603·15-s + (−0.922 − 0.385i)16-s + (0.485 + 0.280i)17-s + (0.929 + 0.350i)18-s + (−0.653 − 1.13i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 + 0.582i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.812 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.83409 - 0.910437i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.83409 - 0.910437i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.54 + 1.26i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.126 - 0.219i)T + (-4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (-1.78 - 3.09i)T + (-12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (-6.82 - 3.94i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 18.1T + 169T^{2} \) |
| 17 | \( 1 + (-8.26 - 4.76i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (12.4 + 21.4i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-2.14 - 3.72i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 28.3iT - 841T^{2} \) |
| 31 | \( 1 + (-28.2 - 16.3i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (25.9 - 14.9i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 45.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 24.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (44.0 - 25.4i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-54.3 - 31.4i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-37.0 + 64.0i)T + (-1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (25.2 + 43.8i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (108. + 62.7i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 5.33T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-23.6 - 13.6i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-51.5 - 89.2i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 51.5T + 6.88e3T^{2} \) |
| 89 | \( 1 + (133. - 76.9i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 47.0iT - 9.40e3T^{2} \) |
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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
−10.95765776175706756753160125641, −10.40289257494617638774228315492, −9.521529843545190992646732546179, −8.293826481106205935046681588006, −6.81267265349772000667770819449, −6.21381106124386648018945309149, −4.93215181991476219034759020964, −3.96892464956146600193244636287, −2.69975706525275923404549768433, −1.44020616127800215053596811168,
1.37579629487412852054411226636, 3.40740346274174472519756109564, 4.20091124702960508738860475338, 5.56460201893020680788333735823, 6.24598239801672387516455739210, 7.17877088275001850090433418608, 8.510187831840753092831854099709, 8.968223820070851468456740336835, 10.28760436934517622638519924697, 11.52028078693300131809460073399