L(s) = 1 | + 24.1i·2-s + (39.5 + 70.6i)3-s − 326.·4-s + (−1.70e3 + 955. i)6-s − 1.04e3·7-s − 1.70e3i·8-s + (−3.42e3 + 5.59e3i)9-s + 1.95e4i·11-s + (−1.29e4 − 2.30e4i)12-s − 2.90e4·13-s − 2.51e4i·14-s − 4.24e4·16-s − 1.22e5i·17-s + (−1.35e5 − 8.27e4i)18-s + 1.89e5·19-s + ⋯ |
L(s) = 1 | + 1.50i·2-s + (0.488 + 0.872i)3-s − 1.27·4-s + (−1.31 + 0.737i)6-s − 0.434·7-s − 0.415i·8-s + (−0.522 + 0.852i)9-s + 1.33i·11-s + (−0.623 − 1.11i)12-s − 1.01·13-s − 0.654i·14-s − 0.648·16-s − 1.46i·17-s + (−1.28 − 0.788i)18-s + 1.45·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.814084 - 0.477171i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.814084 - 0.477171i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-39.5 - 70.6i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 24.1iT - 256T^{2} \) |
| 7 | \( 1 + 1.04e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 1.95e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 2.90e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.22e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.89e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 1.12e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 1.08e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.19e6T + 8.52e11T^{2} \) |
| 37 | \( 1 - 2.84e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 3.90e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 8.64e5T + 1.16e13T^{2} \) |
| 47 | \( 1 + 1.48e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 3.65e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 1.46e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.60e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 2.14e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 4.10e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 4.70e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 2.43e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 6.88e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 3.39e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.50e7T + 7.83e15T^{2} \) |
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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
−14.40270587609812348531970281808, −13.14268491431122164523674547676, −11.57919045274372174076206070477, −9.767874183231186231749346561272, −9.344026743510080301819253183432, −7.76716483197012441629860801711, −7.04818960380263598455514323404, −5.35018258046321228803402420584, −4.54208398291992617369780288131, −2.68201754407042002175106789942,
0.27127599434714089752330019607, 1.47174422280377964203551797655, 2.79105741453478080934520821192, 3.69435847269121319944657941510, 5.86800514547756419737083620796, 7.44070694054519919206061913722, 8.813130076451999247301351793608, 9.799436820073931853079905381237, 11.08406786671001269896016309357, 12.01530577182918188843884098616