| L(s) = 1 | + (0.249 + 1.71i)3-s + (−1.47 + 1.75i)5-s + (0.590 − 1.62i)7-s + (−2.87 + 0.854i)9-s + (−0.867 + 0.728i)11-s + (−1.22 + 6.94i)13-s + (−3.38 − 2.09i)15-s + (−1.48 − 0.857i)17-s + (3.94 − 2.27i)19-s + (2.92 + 0.608i)21-s + (−6.51 + 2.37i)23-s + (−0.0475 − 0.269i)25-s + (−2.18 − 4.71i)27-s + (−6.35 + 1.12i)29-s + (−1.56 − 4.30i)31-s + ⋯ |
| L(s) = 1 | + (0.143 + 0.989i)3-s + (−0.660 + 0.786i)5-s + (0.223 − 0.613i)7-s + (−0.958 + 0.284i)9-s + (−0.261 + 0.219i)11-s + (−0.339 + 1.92i)13-s + (−0.873 − 0.540i)15-s + (−0.360 − 0.207i)17-s + (0.904 − 0.522i)19-s + (0.639 + 0.132i)21-s + (−1.35 + 0.494i)23-s + (−0.00950 − 0.0538i)25-s + (−0.419 − 0.907i)27-s + (−1.17 + 0.208i)29-s + (−0.281 − 0.773i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 - 0.527i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.260824 + 0.913957i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.260824 + 0.913957i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.249 - 1.71i)T \) |
| good | 5 | \( 1 + (1.47 - 1.75i)T + (-0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.590 + 1.62i)T + (-5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (0.867 - 0.728i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (1.22 - 6.94i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (1.48 + 0.857i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.94 + 2.27i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.51 - 2.37i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (6.35 - 1.12i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (1.56 + 4.30i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-1.47 + 2.55i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.14 - 1.25i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.337 - 0.402i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-8.97 - 3.26i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 8.57iT - 53T^{2} \) |
| 59 | \( 1 + (-11.0 - 9.25i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.269 + 0.0979i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-5.85 - 1.03i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (1.36 - 2.36i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (1.58 + 2.74i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.41 + 1.65i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (0.731 + 4.14i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (5.83 - 3.36i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.21 - 2.69i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.32306733463016500146424815314, −10.78116417873348651299609499866, −9.636665696990484654981868978957, −9.118208214648899951083261024632, −7.66294954810227016654344322383, −7.14600948156957838897011356054, −5.74422306379747635888204060208, −4.35964452013942080793406141090, −3.87321010254664929864932239092, −2.41592029664698926080371756660,
0.57887215987334659179598716001, 2.30617181039407182548157520253, 3.62779569119362482047988939359, 5.26314957371030454833940929279, 5.86456625247737370737543616128, 7.34419898502291063546949442908, 8.152024954105793966756159845358, 8.502029782517688287497650748150, 9.812395646452195012806136785030, 10.98362442503651308857106492358
