| L(s) = 1 | + (−1.94 + 5.97i)2-s + (9.69 + 7.04i)3-s + (−6.00 − 4.36i)4-s + (46.5 + 30.8i)5-s + (−60.8 + 44.2i)6-s − 78.1·7-s + (−124. + 90.6i)8-s + (−30.7 − 94.6i)9-s + (−274. + 218. i)10-s + (35.4 − 109. i)11-s + (−27.4 − 84.5i)12-s + (239. + 735. i)13-s + (151. − 466. i)14-s + (234. + 627. i)15-s + (−372. − 1.14e3i)16-s + (1.74e3 − 1.26e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.343 + 1.05i)2-s + (0.621 + 0.451i)3-s + (−0.187 − 0.136i)4-s + (0.833 + 0.552i)5-s + (−0.690 + 0.501i)6-s − 0.602·7-s + (−0.689 + 0.501i)8-s + (−0.126 − 0.389i)9-s + (−0.869 + 0.690i)10-s + (0.0883 − 0.272i)11-s + (−0.0551 − 0.169i)12-s + (0.392 + 1.20i)13-s + (0.206 − 0.636i)14-s + (0.268 + 0.719i)15-s + (−0.364 − 1.12i)16-s + (1.46 − 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 - 0.795i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.712701 + 1.43926i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.712701 + 1.43926i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-46.5 - 30.8i)T \) |
| good | 2 | \( 1 + (1.94 - 5.97i)T + (-25.8 - 18.8i)T^{2} \) |
| 3 | \( 1 + (-9.69 - 7.04i)T + (75.0 + 231. i)T^{2} \) |
| 7 | \( 1 + 78.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-35.4 + 109. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-239. - 735. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-1.74e3 + 1.26e3i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-391. + 284. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (252. - 778. i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-508. - 369. i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-568. + 413. i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-1.06e3 - 3.26e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (4.38e3 + 1.34e4i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 2.19e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.87e4 + 1.36e4i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-6.52e3 - 4.74e3i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (2.61e3 + 8.04e3i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (9.07e3 - 2.79e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (2.41e4 - 1.75e4i)T + (4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (4.53e4 + 3.29e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (4.77e3 - 1.46e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (3.08e4 + 2.24e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (6.30e4 - 4.57e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-2.85e4 + 8.77e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (6.70e4 + 4.87e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−16.72560634920900327121925399549, −15.84790691554325137914521759093, −14.56776447386747301862853519713, −13.87111544296674216549647917946, −11.77729480708661059534866276815, −9.779373907329111360282228318955, −8.928018780040817152232268560574, −7.12672573241459599897919125779, −5.91494493922488328484449702302, −3.08766917086965864161309063261,
1.36135416145434641315853484844, 2.98787364400400591622576066857, 5.93361116234322872616681795359, 8.160782822018314471925643545430, 9.621154405216007667106279727948, 10.57557937430362110341776542146, 12.47555429767733699796160685195, 13.10884005787922230462288972725, 14.58786671286407520032876061431, 16.25380968910347029854890815353
