L(s) = 1 | + (−1.16 − 1.27i)3-s + (0.898 + 0.753i)5-s + (−1.31 + 2.29i)7-s + (−0.267 + 2.98i)9-s + (−2.05 − 2.45i)11-s + (1.12 − 1.33i)13-s + (−0.0867 − 2.02i)15-s + (−1.58 − 2.74i)17-s + (−5.76 − 3.32i)19-s + (4.47 − 0.997i)21-s + (−1.94 − 5.34i)23-s + (−0.629 − 3.56i)25-s + (4.13 − 3.15i)27-s + (3.84 + 4.58i)29-s + (−2.86 + 3.41i)31-s + ⋯ |
L(s) = 1 | + (−0.674 − 0.737i)3-s + (0.401 + 0.337i)5-s + (−0.498 + 0.867i)7-s + (−0.0890 + 0.996i)9-s + (−0.620 − 0.739i)11-s + (0.311 − 0.371i)13-s + (−0.0223 − 0.524i)15-s + (−0.383 − 0.664i)17-s + (−1.32 − 0.763i)19-s + (0.976 − 0.217i)21-s + (−0.405 − 1.11i)23-s + (−0.125 − 0.713i)25-s + (0.795 − 0.606i)27-s + (0.714 + 0.851i)29-s + (−0.514 + 0.612i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 + 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.950 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0583979 - 0.367534i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0583979 - 0.367534i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.16 + 1.27i)T \) |
| 7 | \( 1 + (1.31 - 2.29i)T \) |
good | 5 | \( 1 + (-0.898 - 0.753i)T + (0.868 + 4.92i)T^{2} \) |
| 11 | \( 1 + (2.05 + 2.45i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.12 + 1.33i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.58 + 2.74i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.76 + 3.32i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.94 + 5.34i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-3.84 - 4.58i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (2.86 - 3.41i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + 9.50T + 37T^{2} \) |
| 41 | \( 1 + (7.74 + 6.50i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.80 - 1.38i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (6.48 - 5.44i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-11.7 - 6.78i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.90 + 10.7i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-1.66 - 1.98i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (14.4 - 5.25i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-2.08 - 1.20i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 1.16iT - 73T^{2} \) |
| 79 | \( 1 + (9.42 + 3.43i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-5.29 + 4.44i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-5.74 + 9.95i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.0688 + 0.189i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−10.40346626578271850047661482636, −8.795574266162310537894490747896, −8.450546254385770433475089729138, −7.04938686684068835847787793788, −6.42037772626222071327752830518, −5.68565205470378019225089719803, −4.78434610465111681440422966199, −2.98957539209261960878142674499, −2.14560940111597181122359248354, −0.19320445863608968163833612452,
1.76385161095677844703426698171, 3.62553111173389522136855193046, 4.31169756397784126290560929298, 5.36778142188733466801531118070, 6.24403074610601637930970710096, 7.06688554006874886135080532093, 8.264359679065482211512229879101, 9.256673136097833386714997429645, 10.17214042361814710769513974267, 10.37184035231106047544306957235