| L(s) = 1 | + (1.01 + 2.17i)2-s + (0.586 + 1.62i)3-s + (−2.42 + 2.89i)4-s + (−2.00 − 0.988i)5-s + (−2.95 + 2.93i)6-s + (0.104 + 0.388i)7-s + (−4.12 − 1.10i)8-s + (−2.31 + 1.91i)9-s + (0.117 − 5.37i)10-s + (−0.205 + 0.118i)11-s + (−6.13 − 2.26i)12-s + (4.48 − 3.13i)13-s + (−0.741 + 0.622i)14-s + (0.436 − 3.84i)15-s + (−0.470 − 2.67i)16-s + (1.88 + 4.04i)17-s + ⋯ |
| L(s) = 1 | + (0.718 + 1.54i)2-s + (0.338 + 0.941i)3-s + (−1.21 + 1.44i)4-s + (−0.896 − 0.442i)5-s + (−1.20 + 1.19i)6-s + (0.0393 + 0.147i)7-s + (−1.45 − 0.390i)8-s + (−0.771 + 0.636i)9-s + (0.0370 − 1.69i)10-s + (−0.0618 + 0.0357i)11-s + (−1.77 − 0.652i)12-s + (1.24 − 0.870i)13-s + (−0.198 + 0.166i)14-s + (0.112 − 0.993i)15-s + (−0.117 − 0.667i)16-s + (0.457 + 0.981i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0994815 - 1.64237i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0994815 - 1.64237i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.586 - 1.62i)T \) |
| 5 | \( 1 + (2.00 + 0.988i)T \) |
| 19 | \( 1 + (-0.558 - 4.32i)T \) |
| good | 2 | \( 1 + (-1.01 - 2.17i)T + (-1.28 + 1.53i)T^{2} \) |
| 7 | \( 1 + (-0.104 - 0.388i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (0.205 - 0.118i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.48 + 3.13i)T + (4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (-1.88 - 4.04i)T + (-10.9 + 13.0i)T^{2} \) |
| 23 | \( 1 + (0.128 - 0.0112i)T + (22.6 - 3.99i)T^{2} \) |
| 29 | \( 1 + (3.78 - 1.37i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-4.23 + 7.32i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.80 + 5.80i)T - 37iT^{2} \) |
| 41 | \( 1 + (-7.51 + 1.32i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (5.29 + 0.463i)T + (42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (7.11 + 3.31i)T + (30.2 + 36.0i)T^{2} \) |
| 53 | \( 1 + (-0.604 + 0.0529i)T + (52.1 - 9.20i)T^{2} \) |
| 59 | \( 1 + (3.38 + 1.23i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-4.41 - 3.70i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (0.495 - 1.06i)T + (-43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (-0.164 - 0.196i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (1.25 - 1.78i)T + (-24.9 - 68.5i)T^{2} \) |
| 79 | \( 1 + (-9.95 + 1.75i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (2.48 + 9.27i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (0.817 - 4.63i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (5.84 - 2.72i)T + (62.3 - 74.3i)T^{2} \) |
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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
−12.71029008439863283318077201375, −11.44699796178266326864384238570, −10.37770157504840635341482870023, −9.006193499347787672002850732399, −8.125489771474857935927898669413, −7.77604958725527865825597221983, −6.06234212216656559469432013069, −5.38962857852659063772338351460, −4.12567455759131274961512773822, −3.58000022237810306686055719797,
1.08477033815796044610822966968, 2.68680673396214659316072919810, 3.54478914619450714114220153857, 4.72527215217593910303939157698, 6.34823826828205809754094934973, 7.41672271198503809117891696236, 8.604128601819526430729415325436, 9.636502780460096584524437579196, 11.03907703309820193889647979072, 11.42464151651220667594779464163
