| L(s) = 1 | + (1.08 + 2.10i)2-s + (−1.11 + 3.12i)3-s + (−2.07 + 2.93i)4-s + (−0.863 − 0.375i)5-s + (−7.77 + 1.06i)6-s + (−2.24 − 1.36i)7-s + (−3.73 − 0.514i)8-s + (−6.22 − 5.06i)9-s + (−0.151 − 2.22i)10-s + (2.29 − 0.642i)11-s + (−6.88 − 9.75i)12-s + (1.36 + 6.55i)13-s + (0.423 − 6.18i)14-s + (2.13 − 2.28i)15-s + (−0.582 − 1.63i)16-s + (0.962 + 0.269i)17-s + ⋯ |
| L(s) = 1 | + (0.769 + 1.48i)2-s + (−0.641 + 1.80i)3-s + (−1.03 + 1.46i)4-s + (−0.386 − 0.167i)5-s + (−3.17 + 0.436i)6-s + (−0.846 − 0.515i)7-s + (−1.32 − 0.181i)8-s + (−2.07 − 1.68i)9-s + (−0.0480 − 0.702i)10-s + (0.691 − 0.193i)11-s + (−1.98 − 2.81i)12-s + (0.378 + 1.81i)13-s + (0.113 − 1.65i)14-s + (0.550 − 0.589i)15-s + (−0.145 − 0.409i)16-s + (0.233 + 0.0654i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.747i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.664 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.790563 - 0.355052i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.790563 - 0.355052i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-0.962 - 0.269i)T \) |
| 47 | \( 1 + (3.64 - 5.80i)T \) |
| good | 2 | \( 1 + (-1.08 - 2.10i)T + (-1.15 + 1.63i)T^{2} \) |
| 3 | \( 1 + (1.11 - 3.12i)T + (-2.32 - 1.89i)T^{2} \) |
| 5 | \( 1 + (0.863 + 0.375i)T + (3.41 + 3.65i)T^{2} \) |
| 7 | \( 1 + (2.24 + 1.36i)T + (3.22 + 6.21i)T^{2} \) |
| 11 | \( 1 + (-2.29 + 0.642i)T + (9.39 - 5.71i)T^{2} \) |
| 13 | \( 1 + (-1.36 - 6.55i)T + (-11.9 + 5.17i)T^{2} \) |
| 19 | \( 1 + (0.293 - 0.127i)T + (12.9 - 13.8i)T^{2} \) |
| 23 | \( 1 + (2.13 - 4.11i)T + (-13.2 - 18.7i)T^{2} \) |
| 29 | \( 1 + (0.518 - 2.49i)T + (-26.5 - 11.5i)T^{2} \) |
| 31 | \( 1 + (1.93 + 5.44i)T + (-24.0 + 19.5i)T^{2} \) |
| 37 | \( 1 + (0.675 + 9.87i)T + (-36.6 + 5.03i)T^{2} \) |
| 41 | \( 1 + (-3.32 + 0.456i)T + (39.4 - 11.0i)T^{2} \) |
| 43 | \( 1 + (4.62 - 6.54i)T + (-14.3 - 40.5i)T^{2} \) |
| 53 | \( 1 + (-2.21 + 0.303i)T + (51.0 - 14.2i)T^{2} \) |
| 59 | \( 1 + (-0.0280 - 0.0396i)T + (-19.7 + 55.5i)T^{2} \) |
| 61 | \( 1 + (0.491 - 7.17i)T + (-60.4 - 8.30i)T^{2} \) |
| 67 | \( 1 + (-7.82 + 4.76i)T + (30.8 - 59.4i)T^{2} \) |
| 71 | \( 1 + (2.26 - 4.36i)T + (-40.9 - 58.0i)T^{2} \) |
| 73 | \( 1 + (-0.639 + 0.519i)T + (14.8 - 71.4i)T^{2} \) |
| 79 | \( 1 + (6.21 - 6.65i)T + (-5.39 - 78.8i)T^{2} \) |
| 83 | \( 1 + (10.3 - 2.89i)T + (70.9 - 43.1i)T^{2} \) |
| 89 | \( 1 + (-9.34 - 4.05i)T + (60.7 + 65.0i)T^{2} \) |
| 97 | \( 1 + (-1.42 + 4.02i)T + (-75.2 - 61.2i)T^{2} \) |
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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
−11.11175076742160004774732390045, −9.858314780998736813980287036373, −9.314708816072390568903476138315, −8.526455568606965444305996218392, −7.23277218500473214650844958358, −6.29493571553040839009246047622, −5.83501287737342645763689305184, −4.68071425011040415612059928631, −3.93715094776789235284130367195, −3.70437242324426746494395561652,
0.37008107944800345956624494298, 1.57549212438868535502350993719, 2.73697886895434149437210482985, 3.47503589227349351930364982845, 5.12892648875690447208063858578, 5.89055218140279701299847963850, 6.68150942073329014499633440734, 7.74356009164063635601975973548, 8.636209812641409376738880862969, 10.00302354925739505873829417748
