| L(s) = 1 | + (13.5 + 7.79i)3-s + (112. − 64.6i)5-s + (−289. − 183. i)7-s + (121.5 + 210. i)9-s + (356. − 616. i)11-s + 3.36e3i·13-s + 2.01e3·15-s + (−4.85e3 − 2.80e3i)17-s + (−4.64e3 + 2.68e3i)19-s + (−2.48e3 − 4.73e3i)21-s + (−3.02e3 − 5.23e3i)23-s + (550. − 953. i)25-s + 3.78e3i·27-s − 2.57e4·29-s + (1.71e4 + 9.91e3i)31-s + ⋯ |
| L(s) = 1 | + (0.5 + 0.288i)3-s + (0.896 − 0.517i)5-s + (−0.844 − 0.534i)7-s + (0.166 + 0.288i)9-s + (0.267 − 0.463i)11-s + 1.53i·13-s + 0.597·15-s + (−0.988 − 0.570i)17-s + (−0.677 + 0.391i)19-s + (−0.268 − 0.511i)21-s + (−0.248 − 0.430i)23-s + (0.0352 − 0.0609i)25-s + 0.192i·27-s − 1.05·29-s + (0.576 + 0.332i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.572 - 0.819i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.572 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.182850398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.182850398\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-13.5 - 7.79i)T \) |
| 7 | \( 1 + (289. + 183. i)T \) |
| good | 5 | \( 1 + (-112. + 64.6i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-356. + 616. i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 - 3.36e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (4.85e3 + 2.80e3i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (4.64e3 - 2.68e3i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (3.02e3 + 5.23e3i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + 2.57e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + (-1.71e4 - 9.91e3i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-3.16e4 - 5.48e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 - 9.84e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 3.85e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-4.84e4 + 2.79e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (7.68e4 - 1.33e5i)T + (-1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (5.44e4 + 3.14e4i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-2.04e5 + 1.17e5i)T + (2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-1.19e5 + 2.06e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + 4.49e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (-2.72e5 - 1.57e5i)T + (7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-7.37e4 - 1.27e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 - 4.42e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (-4.79e5 + 2.76e5i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 - 1.66e6iT - 8.32e11T^{2} \) |
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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
−10.70720220359146008543224933269, −9.552779956280966289582329512190, −9.319324020570490223153025676485, −8.263431659751194598494839181941, −6.82160841803709830522753590622, −6.19705568376137822228368242251, −4.74480033267127381406461875658, −3.86132616730644564420557029257, −2.49816033127122959342062626293, −1.37518818649318321460285429385,
0.23229359275303344690113230063, 1.98867754208312486577102792644, 2.70052782358636742619439409288, 3.90431689868228858520219173851, 5.58693293339999477885718644093, 6.30286680974792940295733737908, 7.24906512548881679234721433278, 8.435312327663193541483061072045, 9.346325346413316944569707078344, 10.08803979133291007110196880632
