L(s) = 1 | − 29.2i·2-s + (80.0 + 12.3i)3-s − 596.·4-s + (360. − 2.33e3i)6-s − 2.07e3·7-s + 9.94e3i·8-s + (6.25e3 + 1.97e3i)9-s + 1.19e4i·11-s + (−4.77e4 − 7.37e3i)12-s − 2.78e4·13-s + 6.05e4i·14-s + 1.37e5·16-s + 3.24e4i·17-s + (5.77e4 − 1.82e5i)18-s + 1.22e5·19-s + ⋯ |
L(s) = 1 | − 1.82i·2-s + (0.988 + 0.152i)3-s − 2.33·4-s + (0.278 − 1.80i)6-s − 0.863·7-s + 2.42i·8-s + (0.953 + 0.301i)9-s + 0.817i·11-s + (−2.30 − 0.355i)12-s − 0.973·13-s + 1.57i·14-s + 2.10·16-s + 0.388i·17-s + (0.550 − 1.74i)18-s + 0.943·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.152i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.988 + 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(1.34758 - 0.103417i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.34758 - 0.103417i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-80.0 - 12.3i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 29.2iT - 256T^{2} \) |
| 7 | \( 1 + 2.07e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 1.19e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 2.78e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 3.24e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.22e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 2.66e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 7.93e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 8.76e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 1.82e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 1.49e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 1.65e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 3.35e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 2.01e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 2.16e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.66e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 2.51e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 4.99e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 4.82e6T + 8.06e14T^{2} \) |
| 79 | \( 1 + 2.56e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 4.83e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 3.21e6iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 5.09e7T + 7.83e15T^{2} \) |
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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
−12.71204619633392063507237020860, −11.92540907589542397553677139463, −10.33785054436412205265802235760, −9.756096344806845908198172379805, −8.923860914293555345555569972091, −7.36338273803259931596047794086, −4.86325183559149456353104620128, −3.58771517949170652848444911335, −2.67786152408593906868975791471, −1.44624323598003177835797704541,
0.38429903719803325911130696861, 3.05570181712684594703639343264, 4.58325796482545656220078355326, 6.10297462457138885024759077488, 7.13361491433545669281152653278, 8.059063768928530595967913384287, 9.120679045845759844952056317248, 9.932733613823099799576726085986, 12.37746970867478209586113098268, 13.56168316140343595615565282697