| L(s) = 1 | + (63.1 + 50.7i)3-s + (−94.4 − 617. i)5-s − 2.12e3i·7-s + (1.41e3 + 6.40e3i)9-s + 1.88e4i·11-s − 1.61e3i·13-s + (2.53e4 − 4.38e4i)15-s + 1.28e5·17-s − 1.23e5·19-s + (1.07e5 − 1.33e5i)21-s − 4.63e5·23-s + (−3.72e5 + 1.16e5i)25-s + (−2.35e5 + 4.76e5i)27-s + 8.67e5i·29-s + 1.29e6·31-s + ⋯ |
| L(s) = 1 | + (0.779 + 0.626i)3-s + (−0.151 − 0.988i)5-s − 0.883i·7-s + (0.215 + 0.976i)9-s + 1.28i·11-s − 0.0564i·13-s + (0.501 − 0.865i)15-s + 1.53·17-s − 0.943·19-s + (0.553 − 0.688i)21-s − 1.65·23-s + (−0.954 + 0.298i)25-s + (−0.443 + 0.896i)27-s + 1.22i·29-s + 1.39·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.501 - 0.865i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.501 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.567139643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.567139643\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-63.1 - 50.7i)T \) |
| 5 | \( 1 + (94.4 + 617. i)T \) |
| good | 7 | \( 1 + 2.12e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 1.88e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 1.61e3iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 1.28e5T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.23e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 4.63e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 8.67e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.29e6T + 8.52e11T^{2} \) |
| 37 | \( 1 + 1.60e5iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 4.10e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 1.57e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 5.70e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 8.30e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 1.28e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 2.79e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 2.02e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 2.19e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 3.05e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 6.31e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 3.53e6T + 2.25e15T^{2} \) |
| 89 | \( 1 + 9.26e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.20e8iT - 7.83e15T^{2} \) |
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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
−10.33511808989096710352073331868, −10.04627874980435637455231312032, −8.906778217955979018923096904798, −7.998834540669336305145887013145, −7.23456703413733172281138366844, −5.49707077393629786359230079493, −4.38879216646601419564949464565, −3.83346221692392887216491941808, −2.21802848651575790387862669060, −1.03951032738473970125050512818,
0.55564645012943610033021127629, 2.09836608910772631638923138454, 2.93952016514648494336095578049, 3.87185725044317364669734635315, 5.89824331320847427222274045139, 6.39790418324256882139616174689, 7.88795185915922693224653121171, 8.252209882939468443231282884206, 9.517831154954536647425736855462, 10.45022970630337832307620575616
