| L(s) = 1 | + (−2.82 − 4.89i)2-s + (−12.7 − 7.36i)3-s + (−15.9 + 27.7i)4-s + (−106. + 61.6i)5-s + 83.3i·6-s + (−309. + 147. i)7-s + 181.·8-s + (−255. − 443. i)9-s + (603. + 348. i)10-s + (292. − 506. i)11-s + (408. − 235. i)12-s − 1.72e3i·13-s + (1.59e3 + 1.10e3i)14-s + 1.81e3·15-s + (−512. − 886. i)16-s + (1.07e3 + 621. i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.472 − 0.272i)3-s + (−0.249 + 0.433i)4-s + (−0.853 + 0.493i)5-s + 0.385i·6-s + (−0.902 + 0.429i)7-s + 0.353·8-s + (−0.351 − 0.608i)9-s + (0.603 + 0.348i)10-s + (0.219 − 0.380i)11-s + (0.236 − 0.136i)12-s − 0.784i·13-s + (0.582 + 0.400i)14-s + 0.538·15-s + (−0.125 − 0.216i)16-s + (0.219 + 0.126i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0213781 + 0.0881033i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0213781 + 0.0881033i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.82 + 4.89i)T \) |
| 7 | \( 1 + (309. - 147. i)T \) |
| good | 3 | \( 1 + (12.7 + 7.36i)T + (364.5 + 631. i)T^{2} \) |
| 5 | \( 1 + (106. - 61.6i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-292. + 506. i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + 1.72e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (-1.07e3 - 621. i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (1.00e4 - 5.80e3i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (3.91e3 + 6.77e3i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 - 4.02e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + (-8.27e3 - 4.77e3i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (4.18e4 + 7.24e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 - 6.65e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.37e5T + 6.32e9T^{2} \) |
| 47 | \( 1 + (1.22e5 - 7.07e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-1.01e5 + 1.76e5i)T + (-1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (7.92e3 + 4.57e3i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (2.80e4 - 1.61e4i)T + (2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-1.73e4 + 3.01e4i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + 7.64e4T + 1.28e11T^{2} \) |
| 73 | \( 1 + (5.48e5 + 3.16e5i)T + (7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-2.40e5 - 4.16e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + 7.74e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (-3.62e5 + 2.09e5i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 - 1.26e6iT - 8.32e11T^{2} \) |
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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
−17.78192851038680170798461766783, −16.33041874567845049461991253493, −14.85225864579734450536296725002, −12.72682998276238035833259168178, −11.78346222351975706418705403592, −10.33718048605998111646189235511, −8.440769700455981893589758909666, −6.40064284635542909788011158859, −3.35840157923930725669284949627, −0.07782869679193240682227583333,
4.54020170766729678799218717100, 6.69292121467620542601053896388, 8.479157816787994468742971312205, 10.21960479920297810840365172891, 11.88252757810993459122193536281, 13.62671841158766461222619411286, 15.45729253552894182125772226203, 16.42606446257303708896868432688, 17.25905188244513090761368218889, 19.18786123942384851543416216762
