| 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
−19.18786123942384851543416216762, −17.25905188244513090761368218889, −16.42606446257303708896868432688, −15.45729253552894182125772226203, −13.62671841158766461222619411286, −11.88252757810993459122193536281, −10.21960479920297810840365172891, −8.479157816787994468742971312205, −6.69292121467620542601053896388, −4.54020170766729678799218717100,
0.07782869679193240682227583333, 3.35840157923930725669284949627, 6.40064284635542909788011158859, 8.440769700455981893589758909666, 10.33718048605998111646189235511, 11.78346222351975706418705403592, 12.72682998276238035833259168178, 14.85225864579734450536296725002, 16.33041874567845049461991253493, 17.78192851038680170798461766783
