| L(s) = 1 | + (−0.412 + 0.308i)3-s + (−1.14 + 1.91i)5-s + (0.218 − 3.05i)7-s + (−0.770 + 2.62i)9-s + (−1.30 + 2.03i)11-s + (−0.154 + 0.0110i)13-s + (−0.119 − 1.14i)15-s + (−1.84 + 4.94i)17-s + (−0.917 + 2.00i)19-s + (0.852 + 1.32i)21-s + (−4.48 + 1.71i)23-s + (−2.36 − 4.40i)25-s + (−1.03 − 2.76i)27-s + (−3.67 + 1.67i)29-s + (0.161 + 1.12i)31-s + ⋯ |
| L(s) = 1 | + (−0.238 + 0.178i)3-s + (−0.513 + 0.858i)5-s + (0.0825 − 1.15i)7-s + (−0.256 + 0.874i)9-s + (−0.394 + 0.613i)11-s + (−0.0429 + 0.00307i)13-s + (−0.0307 − 0.295i)15-s + (−0.447 + 1.20i)17-s + (−0.210 + 0.460i)19-s + (0.186 + 0.289i)21-s + (−0.934 + 0.356i)23-s + (−0.473 − 0.880i)25-s + (−0.198 − 0.532i)27-s + (−0.681 + 0.311i)29-s + (0.0289 + 0.201i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.693 - 0.720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.268447 + 0.630426i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.268447 + 0.630426i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.14 - 1.91i)T \) |
| 23 | \( 1 + (4.48 - 1.71i)T \) |
| good | 3 | \( 1 + (0.412 - 0.308i)T + (0.845 - 2.87i)T^{2} \) |
| 7 | \( 1 + (-0.218 + 3.05i)T + (-6.92 - 0.996i)T^{2} \) |
| 11 | \( 1 + (1.30 - 2.03i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (0.154 - 0.0110i)T + (12.8 - 1.85i)T^{2} \) |
| 17 | \( 1 + (1.84 - 4.94i)T + (-12.8 - 11.1i)T^{2} \) |
| 19 | \( 1 + (0.917 - 2.00i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (3.67 - 1.67i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.161 - 1.12i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-0.622 - 1.14i)T + (-20.0 + 31.1i)T^{2} \) |
| 41 | \( 1 + (11.6 - 3.41i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-3.54 - 4.73i)T + (-12.1 + 41.2i)T^{2} \) |
| 47 | \( 1 + (-6.85 + 6.85i)T - 47iT^{2} \) |
| 53 | \( 1 + (-10.4 - 0.749i)T + (52.4 + 7.54i)T^{2} \) |
| 59 | \( 1 + (-7.94 - 6.88i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (3.34 - 0.481i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-4.21 - 0.916i)T + (60.9 + 27.8i)T^{2} \) |
| 71 | \( 1 + (5.76 - 3.70i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-4.68 + 1.74i)T + (55.1 - 47.8i)T^{2} \) |
| 79 | \( 1 + (-8.94 + 10.3i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-2.49 + 1.36i)T + (44.8 - 69.8i)T^{2} \) |
| 89 | \( 1 + (-0.438 + 3.05i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (14.4 + 7.88i)T + (52.4 + 81.6i)T^{2} \) |
| 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
−11.18994124099011245558626689999, −10.35987941634967939654377510423, −10.18414829502069292528450408956, −8.433725675502619592530792415632, −7.65956692150049380269801001521, −6.94508314200012419106343277667, −5.78434251969379523399850614444, −4.46338671642770515351850039795, −3.66765156252069441078800829883, −2.08134679405727922118653265680,
0.41770421170262192406284471379, 2.41694882861840195403610761996, 3.82409110556776602781236601931, 5.14723104486778097747051629561, 5.82753607461250488442797487534, 7.00049493546351921055449256647, 8.212110609415040554367294463671, 8.895479074201827282902755748290, 9.535018401108301249408649921703, 10.99623977487015789540438932948
