| L(s) = 1 | + (−1.35 + 0.395i)2-s + (0.443 + 0.663i)3-s + (1.68 − 1.07i)4-s + (0.154 + 0.775i)5-s + (−0.864 − 0.726i)6-s + (1.53 + 3.71i)7-s + (−1.86 + 2.12i)8-s + (0.904 − 2.18i)9-s + (−0.516 − 0.992i)10-s + (−1.83 − 1.22i)11-s + (1.46 + 0.644i)12-s + (0.559 − 2.81i)13-s + (−3.55 − 4.43i)14-s + (−0.446 + 0.446i)15-s + (1.69 − 3.62i)16-s + (−2.73 − 2.73i)17-s + ⋯ |
| L(s) = 1 | + (−0.960 + 0.279i)2-s + (0.256 + 0.383i)3-s + (0.843 − 0.536i)4-s + (0.0690 + 0.346i)5-s + (−0.353 − 0.296i)6-s + (0.581 + 1.40i)7-s + (−0.660 + 0.751i)8-s + (0.301 − 0.727i)9-s + (−0.163 − 0.313i)10-s + (−0.553 − 0.369i)11-s + (0.421 + 0.185i)12-s + (0.155 − 0.780i)13-s + (−0.951 − 1.18i)14-s + (−0.115 + 0.115i)15-s + (0.423 − 0.905i)16-s + (−0.663 − 0.663i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.624486 + 0.269894i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.624486 + 0.269894i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.35 - 0.395i)T \) |
| good | 3 | \( 1 + (-0.443 - 0.663i)T + (-1.14 + 2.77i)T^{2} \) |
| 5 | \( 1 + (-0.154 - 0.775i)T + (-4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (-1.53 - 3.71i)T + (-4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (1.83 + 1.22i)T + (4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (-0.559 + 2.81i)T + (-12.0 - 4.97i)T^{2} \) |
| 17 | \( 1 + (2.73 + 2.73i)T + 17iT^{2} \) |
| 19 | \( 1 + (5.68 + 1.13i)T + (17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (-2.55 - 1.05i)T + (16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (2.96 - 1.97i)T + (11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 - 0.201iT - 31T^{2} \) |
| 37 | \( 1 + (8.82 - 1.75i)T + (34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (-7.85 - 3.25i)T + (28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (-1.06 + 1.59i)T + (-16.4 - 39.7i)T^{2} \) |
| 47 | \( 1 + (7.56 + 7.56i)T + 47iT^{2} \) |
| 53 | \( 1 + (-11.1 - 7.47i)T + (20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (1.71 + 8.61i)T + (-54.5 + 22.5i)T^{2} \) |
| 61 | \( 1 + (2.32 + 3.47i)T + (-23.3 + 56.3i)T^{2} \) |
| 67 | \( 1 + (-5.15 - 7.71i)T + (-25.6 + 61.8i)T^{2} \) |
| 71 | \( 1 + (-1.98 - 4.79i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (6.11 - 14.7i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (2.91 - 2.91i)T - 79iT^{2} \) |
| 83 | \( 1 + (0.190 + 0.0378i)T + (76.6 + 31.7i)T^{2} \) |
| 89 | \( 1 + (-0.659 + 0.273i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + 2.80iT - 97T^{2} \) |
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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
−15.27441092471234420438934227491, −14.59027987712970172431805927266, −12.69820198584886534317217125159, −11.40428347497838396715019209702, −10.41256251562888580109082471591, −9.058298827923806884956062236012, −8.416578435513985459949460712546, −6.75675117813126182244992250264, −5.36908776887130168703146356622, −2.66431545392810997934698182345,
1.82897180459988704622301107846, 4.34746877857288807396041038455, 6.85329430332935011495445495000, 7.78546342354545026350753324193, 8.845219920073241940020289801679, 10.45924539692734895060593372593, 10.96173735285192017967851135557, 12.65034668091139347360792105753, 13.46382113287545766138084673310, 14.84479232421954208152628641992
