| L(s) = 1 | + (0.207 + 0.978i)2-s + (−0.637 + 1.96i)3-s + (−0.913 + 0.406i)4-s + (−0.0987 + 0.939i)5-s + (−2.05 − 0.215i)6-s + (−1.08 − 0.979i)7-s + (−0.587 − 0.809i)8-s + (−1.01 − 0.735i)9-s + (−0.939 + 0.0987i)10-s − 2.08i·11-s + (−0.215 − 2.05i)12-s + (3.40 + 5.89i)13-s + (0.732 − 1.26i)14-s + (−1.77 − 0.792i)15-s + (0.669 − 0.743i)16-s + (0.0505 + 0.113i)17-s + ⋯ |
| L(s) = 1 | + (0.147 + 0.691i)2-s + (−0.367 + 1.13i)3-s + (−0.456 + 0.203i)4-s + (−0.0441 + 0.420i)5-s + (−0.837 − 0.0879i)6-s + (−0.411 − 0.370i)7-s + (−0.207 − 0.286i)8-s + (−0.337 − 0.245i)9-s + (−0.297 + 0.0312i)10-s − 0.628i·11-s + (−0.0622 − 0.591i)12-s + (0.943 + 1.63i)13-s + (0.195 − 0.338i)14-s + (−0.459 − 0.204i)15-s + (0.167 − 0.185i)16-s + (0.0122 + 0.0275i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.680 - 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.680 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.380022 + 0.870798i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.380022 + 0.870798i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.207 - 0.978i)T \) |
| 61 | \( 1 + (-1.97 - 7.55i)T \) |
| good | 3 | \( 1 + (0.637 - 1.96i)T + (-2.42 - 1.76i)T^{2} \) |
| 5 | \( 1 + (0.0987 - 0.939i)T + (-4.89 - 1.03i)T^{2} \) |
| 7 | \( 1 + (1.08 + 0.979i)T + (0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + 2.08iT - 11T^{2} \) |
| 13 | \( 1 + (-3.40 - 5.89i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.0505 - 0.113i)T + (-11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (-1.06 - 1.18i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-3.37 + 4.64i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.537 + 0.310i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.22 + 5.74i)T + (-28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (-1.27 + 0.413i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (2.04 + 6.27i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-1.30 + 2.92i)T + (-28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (5.54 - 9.60i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.52 + 4.85i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (1.32 + 6.25i)T + (-53.8 + 23.9i)T^{2} \) |
| 67 | \( 1 + (-4.25 - 0.447i)T + (65.5 + 13.9i)T^{2} \) |
| 71 | \( 1 + (13.1 - 1.38i)T + (69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (0.465 + 4.42i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (5.13 - 11.5i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-13.8 + 2.94i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (12.4 + 4.03i)T + (72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-10.7 - 2.29i)T + (88.6 + 39.4i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.05395696104821026180436659022, −13.06848857825041372863462762695, −11.48346628079178164897175174777, −10.70773994343891406013038743707, −9.590020020556365330358742125802, −8.646554078927316546130438374682, −7.02473326841456953829214779219, −6.05036306189677641709630609734, −4.60527727955997388476881477681, −3.59729610559200305383025213729,
1.20456337162855889767456827673, 3.13281952052362945030300786058, 5.10728553424008034428811311821, 6.29266282763593692926739641967, 7.62260320728694017761055302599, 8.817112437888027577464439871938, 10.07770933344671917112263707217, 11.25828627080243743298816700449, 12.29948662931363386006017300858, 12.91365791703242675137697662642
