| L(s) = 1 | + (−1.40 − 0.196i)2-s + (0.290 + 0.956i)3-s + (1.92 + 0.550i)4-s + (0.0502 + 0.509i)5-s + (−0.218 − 1.39i)6-s + (−1.66 + 2.48i)7-s + (−2.58 − 1.14i)8-s + (−0.831 + 0.555i)9-s + (0.0298 − 0.723i)10-s + (−2.39 + 1.27i)11-s + (0.0316 + 1.99i)12-s + (−0.147 − 0.0145i)13-s + (2.81 − 3.15i)14-s + (−0.473 + 0.196i)15-s + (3.39 + 2.11i)16-s + (−5.02 − 2.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.990 − 0.138i)2-s + (0.167 + 0.552i)3-s + (0.961 + 0.275i)4-s + (0.0224 + 0.227i)5-s + (−0.0892 − 0.570i)6-s + (−0.627 + 0.939i)7-s + (−0.913 − 0.405i)8-s + (−0.277 + 0.185i)9-s + (0.00942 − 0.228i)10-s + (−0.721 + 0.385i)11-s + (0.00914 + 0.577i)12-s + (−0.0408 − 0.00402i)13-s + (0.752 − 0.843i)14-s + (−0.122 + 0.0506i)15-s + (0.848 + 0.528i)16-s + (−1.21 − 0.505i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.128999 + 0.471950i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.128999 + 0.471950i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 + 0.196i)T \) |
| 3 | \( 1 + (-0.290 - 0.956i)T \) |
| good | 5 | \( 1 + (-0.0502 - 0.509i)T + (-4.90 + 0.975i)T^{2} \) |
| 7 | \( 1 + (1.66 - 2.48i)T + (-2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (2.39 - 1.27i)T + (6.11 - 9.14i)T^{2} \) |
| 13 | \( 1 + (0.147 + 0.0145i)T + (12.7 + 2.53i)T^{2} \) |
| 17 | \( 1 + (5.02 + 2.08i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (0.869 - 0.713i)T + (3.70 - 18.6i)T^{2} \) |
| 23 | \( 1 + (0.0792 - 0.398i)T + (-21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-1.28 + 2.41i)T + (-16.1 - 24.1i)T^{2} \) |
| 31 | \( 1 + (3.00 - 3.00i)T - 31iT^{2} \) |
| 37 | \( 1 + (5.49 - 6.69i)T + (-7.21 - 36.2i)T^{2} \) |
| 41 | \( 1 + (7.61 + 1.51i)T + (37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (1.01 - 3.33i)T + (-35.7 - 23.8i)T^{2} \) |
| 47 | \( 1 + (3.94 - 9.53i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-4.05 - 7.57i)T + (-29.4 + 44.0i)T^{2} \) |
| 59 | \( 1 + (-6.40 + 0.630i)T + (57.8 - 11.5i)T^{2} \) |
| 61 | \( 1 + (-9.35 + 2.83i)T + (50.7 - 33.8i)T^{2} \) |
| 67 | \( 1 + (0.0861 - 0.0261i)T + (55.7 - 37.2i)T^{2} \) |
| 71 | \( 1 + (-8.16 - 5.45i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (9.01 + 13.4i)T + (-27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-1.41 - 3.40i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-2.88 - 3.51i)T + (-16.1 + 81.4i)T^{2} \) |
| 89 | \( 1 + (0.951 + 4.78i)T + (-82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (-8.57 + 8.57i)T - 97iT^{2} \) |
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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
−11.47711839137133236922035732494, −10.57409823877671398448069737005, −9.860714816478049768121719924253, −9.004141641226302304216321699726, −8.365452510774664308137981624812, −7.10553604245100448384386735817, −6.21531650874425377690967111725, −4.91562645975025699730019788625, −3.19073839653378927625311093965, −2.29491295567229035873027849227,
0.40327730848165404689044552513, 2.12258536299505336563674545474, 3.55869374488720697171294207372, 5.35644378202535343577145839758, 6.69339529407209279439732468744, 7.10587492589144605563423172739, 8.330682933419327537825847261024, 8.890820913895149774283507643843, 10.12087757489695784659649210364, 10.71750465395939140987798509078
