| L(s) = 1 | + (−5.69 − 3.28i)2-s + (2.79 − 26.8i)3-s + (−10.3 − 17.9i)4-s + (57.4 + 33.1i)5-s + (−104. + 143. i)6-s + (−329. − 93.7i)7-s + 557. i·8-s + (−713. − 149. i)9-s + (−218. − 377. i)10-s + (−603. + 348. i)11-s + (−511. + 228. i)12-s − 824.·13-s + (1.57e3 + 1.61e3i)14-s + (1.05e3 − 1.45e3i)15-s + (1.16e3 − 2.02e3i)16-s + (4.78e3 − 2.76e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.712 − 0.411i)2-s + (0.103 − 0.994i)3-s + (−0.161 − 0.280i)4-s + (0.459 + 0.265i)5-s + (−0.482 + 0.665i)6-s + (−0.961 − 0.273i)7-s + 1.08i·8-s + (−0.978 − 0.205i)9-s + (−0.218 − 0.377i)10-s + (−0.453 + 0.261i)11-s + (−0.295 + 0.132i)12-s − 0.375·13-s + (0.572 + 0.590i)14-s + (0.311 − 0.429i)15-s + (0.285 − 0.494i)16-s + (0.973 − 0.562i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 - 0.311i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0819491 + 0.512724i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0819491 + 0.512724i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.79 + 26.8i)T \) |
| 7 | \( 1 + (329. + 93.7i)T \) |
| good | 2 | \( 1 + (5.69 + 3.28i)T + (32 + 55.4i)T^{2} \) |
| 5 | \( 1 + (-57.4 - 33.1i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 11 | \( 1 + (603. - 348. i)T + (8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + 824.T + 4.82e6T^{2} \) |
| 17 | \( 1 + (-4.78e3 + 2.76e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-2.52e3 + 4.37e3i)T + (-2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (1.72e4 + 9.94e3i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 - 2.34e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + (2.31e4 + 4.01e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-2.41e4 + 4.18e4i)T + (-1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + 1.59e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 2.17e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-3.30e4 - 1.90e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-1.93e5 + 1.11e5i)T + (1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (2.71e5 - 1.56e5i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.26e5 - 2.19e5i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-8.29e4 - 1.43e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + 2.98e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + (2.21e3 + 3.83e3i)T + (-7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-3.10e5 + 5.37e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + 4.14e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (-6.49e5 - 3.75e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 - 2.12e5T + 8.32e11T^{2} \) |
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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
−16.53451924941879166792179430566, −14.50796183867987130598139643266, −13.49849920242713219356101357722, −12.11033115298861499275796891026, −10.41577438381386390486911882503, −9.299386099656732695078415457623, −7.55813185169290351184482676314, −5.87210979957573175319382304998, −2.41224398821020214738173324144, −0.39343914491868736277561447384,
3.53124341014024103188087685339, 5.77611544232410599010687154848, 7.986071668307615889651945298428, 9.412464850148850191815197405064, 10.10548498866010892525274218721, 12.29547174300083318023992530114, 13.75730545469917561803140526446, 15.48580894970293101113400788549, 16.39173462900979940845914117352, 17.19204913581642049935849915510
