| L(s) = 1 | + (−0.573 + 0.819i)2-s + (−0.342 − 0.939i)4-s + (−1.60 + 0.140i)5-s + (1.53 + 1.29i)7-s + (0.965 + 0.258i)8-s + (0.807 − 1.39i)10-s + (−2.64 − 4.57i)11-s + (4.54 + 2.11i)13-s + (−1.94 + 0.520i)14-s + (−0.766 + 0.642i)16-s + (5.07 − 2.36i)17-s + (−0.363 + 0.254i)19-s + (0.682 + 1.46i)20-s + (5.26 + 0.460i)22-s + (1.51 + 5.64i)23-s + ⋯ |
| L(s) = 1 | + (−0.405 + 0.579i)2-s + (−0.171 − 0.469i)4-s + (−0.719 + 0.0629i)5-s + (0.582 + 0.488i)7-s + (0.341 + 0.0915i)8-s + (0.255 − 0.442i)10-s + (−0.797 − 1.38i)11-s + (1.25 + 0.587i)13-s + (−0.518 + 0.139i)14-s + (−0.191 + 0.160i)16-s + (1.23 − 0.574i)17-s + (−0.0833 + 0.0583i)19-s + (0.152 + 0.327i)20-s + (1.12 + 0.0982i)22-s + (0.315 + 1.17i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04232 + 0.454308i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04232 + 0.454308i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.573 - 0.819i)T \) |
| 3 | \( 1 \) |
| 37 | \( 1 + (-6.05 + 0.580i)T \) |
| good | 5 | \( 1 + (1.60 - 0.140i)T + (4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (-1.53 - 1.29i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (2.64 + 4.57i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.54 - 2.11i)T + (8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (-5.07 + 2.36i)T + (10.9 - 13.0i)T^{2} \) |
| 19 | \( 1 + (0.363 - 0.254i)T + (6.49 - 17.8i)T^{2} \) |
| 23 | \( 1 + (-1.51 - 5.64i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (1.91 - 7.15i)T + (-25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (-5.12 + 5.12i)T - 31iT^{2} \) |
| 41 | \( 1 + (-6.41 + 2.33i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-7.63 - 7.63i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.34 - 2.50i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.64 - 9.10i)T + (-9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-0.615 + 7.03i)T + (-58.1 - 10.2i)T^{2} \) |
| 61 | \( 1 + (2.86 - 6.14i)T + (-39.2 - 46.7i)T^{2} \) |
| 67 | \( 1 + (3.96 - 4.72i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.49 + 0.263i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + 8.32iT - 73T^{2} \) |
| 79 | \( 1 + (1.28 + 14.6i)T + (-77.7 + 13.7i)T^{2} \) |
| 83 | \( 1 + (4.01 - 11.0i)T + (-63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (0.608 + 0.0532i)T + (87.6 + 15.4i)T^{2} \) |
| 97 | \( 1 + (10.0 - 2.68i)T + (84.0 - 48.5i)T^{2} \) |
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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
−10.84877985145896988442305156357, −9.511285958643771835357454628536, −8.741957156030589955033686555699, −7.948180685250997754257693549478, −7.45407462531864589182549507183, −5.96627825235429161972062079787, −5.54040192865027909686609806845, −4.15254097643466189492558552509, −2.99987129328961901325738008229, −1.09272854976517038160225154959,
0.961321832650546231222674310079, 2.47158550741128601293313552195, 3.84928402923965866452369072841, 4.54637321486113482716810352301, 5.84687254800053540401531877374, 7.27015852188454677834938335171, 7.946171795841787896489104129032, 8.470278516719319443030062915880, 9.807243663070833676018299315551, 10.45393973012045586017212391839
