| L(s) = 1 | + (13.1 + 7.60i)2-s + (−23.5 + 13.1i)3-s + (83.5 + 144. i)4-s + (−82.1 − 47.4i)5-s + (−410. − 5.43i)6-s + (334. + 73.9i)7-s + 1.56e3i·8-s + (381. − 621. i)9-s + (−721. − 1.24e3i)10-s + (735. − 424. i)11-s + (−3.87e3 − 2.30e3i)12-s + 1.14e3·13-s + (3.84e3 + 3.51e3i)14-s + (2.56e3 + 33.9i)15-s + (−6.56e3 + 1.13e4i)16-s + (−2.38e3 + 1.37e3i)17-s + ⋯ |
| L(s) = 1 | + (1.64 + 0.950i)2-s + (−0.872 + 0.488i)3-s + (1.30 + 2.26i)4-s + (−0.657 − 0.379i)5-s + (−1.90 − 0.0251i)6-s + (0.976 + 0.215i)7-s + 3.06i·8-s + (0.522 − 0.852i)9-s + (−0.721 − 1.24i)10-s + (0.552 − 0.319i)11-s + (−2.24 − 1.33i)12-s + 0.520·13-s + (1.40 + 1.28i)14-s + (0.758 + 0.0100i)15-s + (−1.60 + 2.77i)16-s + (−0.484 + 0.279i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.58664 + 2.28396i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.58664 + 2.28396i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (23.5 - 13.1i)T \) |
| 7 | \( 1 + (-334. - 73.9i)T \) |
| good | 2 | \( 1 + (-13.1 - 7.60i)T + (32 + 55.4i)T^{2} \) |
| 5 | \( 1 + (82.1 + 47.4i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-735. + 424. i)T + (8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 - 1.14e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + (2.38e3 - 1.37e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-3.55e3 + 6.15e3i)T + (-2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (4.81e3 + 2.78e3i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + 2.27e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + (4.26e3 + 7.37e3i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (3.59e4 - 6.22e4i)T + (-1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 - 5.90e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 3.77e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (9.76e4 + 5.63e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-1.75e5 + 1.01e5i)T + (1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-1.50e5 + 8.66e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-2.45e4 + 4.25e4i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (1.56e5 + 2.71e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + 1.00e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + (-1.41e5 - 2.45e5i)T + (-7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (1.86e5 - 3.22e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + 2.43e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (-8.90e5 - 5.13e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 + 1.64e6T + 8.32e11T^{2} \) |
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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
−16.69254190315937193320886942772, −15.71523704658554733290475541588, −14.91811565228472757445771832401, −13.46898013227059878653523928080, −11.94993596819171100742600808191, −11.37907781583403601552140564318, −8.265053829008698900252789065630, −6.52381329258073545136257189685, −5.08490965104130443105968697511, −4.00881389920252961136597994153,
1.56126063629787828727149570808, 4.00532660586457944601758641856, 5.47637981511309625055834103026, 7.12288290249451917432769794651, 10.59747318027983047639021294009, 11.48116769835021597038554998188, 12.23510218605903045336164277077, 13.65122425013430309964114359133, 14.71271503350349739871816376130, 16.04443988806482485598600693540
