| L(s) = 1 | + (−7.22 + 12.5i)2-s + (−5.48 + 3.16i)3-s + (−72.2 − 125. i)4-s + (−169. − 98.1i)5-s − 91.4i·6-s + 1.16e3·8-s + (−344. + 596. i)9-s + (2.45e3 − 1.41e3i)10-s + (388. + 673. i)11-s + (793. + 457. i)12-s − 1.85e3i·13-s + 1.24e3·15-s + (−3.77e3 + 6.52e3i)16-s + (−1.94e3 + 1.12e3i)17-s + (−4.97e3 − 8.61e3i)18-s + (5.36e3 + 3.09e3i)19-s + ⋯ |
| L(s) = 1 | + (−0.902 + 1.56i)2-s + (−0.203 + 0.117i)3-s + (−1.12 − 1.95i)4-s + (−1.35 − 0.784i)5-s − 0.423i·6-s + 2.27·8-s + (−0.472 + 0.818i)9-s + (2.45 − 1.41i)10-s + (0.292 + 0.505i)11-s + (0.458 + 0.264i)12-s − 0.845i·13-s + 0.368·15-s + (−0.920 + 1.59i)16-s + (−0.395 + 0.228i)17-s + (−0.852 − 1.47i)18-s + (0.782 + 0.451i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.445601 + 0.298926i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.445601 + 0.298926i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + (7.22 - 12.5i)T + (-32 - 55.4i)T^{2} \) |
| 3 | \( 1 + (5.48 - 3.16i)T + (364.5 - 631. i)T^{2} \) |
| 5 | \( 1 + (169. + 98.1i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-388. - 673. i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + 1.85e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (1.94e3 - 1.12e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-5.36e3 - 3.09e3i)T + (2.35e7 + 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-662. + 1.14e3i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + 1.95e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + (-4.36e4 + 2.52e4i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-7.03e3 + 1.21e4i)T + (-1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + 6.03e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 5.96e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-1.55e5 - 8.95e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-9.45e4 - 1.63e5i)T + (-1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (9.49e4 - 5.48e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (2.61e5 + 1.50e5i)T + (2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (1.11e5 + 1.93e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 6.67e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (-2.26e5 + 1.30e5i)T + (7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-2.52e5 + 4.38e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + 2.30e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (6.97e5 + 4.02e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 - 4.74e5iT - 8.32e11T^{2} \) |
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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
−15.24195350231483664139492100690, −13.83999046575029069765920545422, −12.22444385708952006934701445036, −10.74470325191842996333602677238, −9.245091249940936317077960522584, −8.085095951644395631586251898028, −7.52366590628703833373025401239, −5.73733252374791965605669205817, −4.50076107528933679129129429281, −0.58795440919935313465446722267,
0.73400365254373646466544149802, 2.91910450556940100829807887484, 3.97162296320349165038852092382, 6.95197555744426837121681731365, 8.364051519773721373643473537320, 9.444892732445628439704085923036, 10.96295522594106581884816314789, 11.57992765565746991348004904134, 12.19516782159455528433879320297, 13.86579332437568088578580603565
