| L(s) = 1 | + (−1.33 + 0.463i)2-s + (−1.31 − 0.876i)3-s + (1.56 − 1.23i)4-s + (3.52 + 0.700i)5-s + (2.15 + 0.562i)6-s + (1.02 − 2.47i)7-s + (−1.52 + 2.38i)8-s + (−0.196 − 0.473i)9-s + (−5.03 + 0.698i)10-s + (1.67 + 2.51i)11-s + (−3.14 + 0.250i)12-s + (−4.41 + 0.878i)13-s + (−0.221 + 3.78i)14-s + (−4.00 − 4.00i)15-s + (0.926 − 3.89i)16-s + (1.12 − 1.12i)17-s + ⋯ |
| L(s) = 1 | + (−0.944 + 0.328i)2-s + (−0.757 − 0.505i)3-s + (0.784 − 0.619i)4-s + (1.57 + 0.313i)5-s + (0.881 + 0.229i)6-s + (0.387 − 0.935i)7-s + (−0.538 + 0.842i)8-s + (−0.0653 − 0.157i)9-s + (−1.59 + 0.220i)10-s + (0.506 + 0.757i)11-s + (−0.907 + 0.0722i)12-s + (−1.22 + 0.243i)13-s + (−0.0591 + 1.01i)14-s + (−1.03 − 1.03i)15-s + (0.231 − 0.972i)16-s + (0.273 − 0.273i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.607177 - 0.0771003i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.607177 - 0.0771003i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.33 - 0.463i)T \) |
| good | 3 | \( 1 + (1.31 + 0.876i)T + (1.14 + 2.77i)T^{2} \) |
| 5 | \( 1 + (-3.52 - 0.700i)T + (4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (-1.02 + 2.47i)T + (-4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-1.67 - 2.51i)T + (-4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (4.41 - 0.878i)T + (12.0 - 4.97i)T^{2} \) |
| 17 | \( 1 + (-1.12 + 1.12i)T - 17iT^{2} \) |
| 19 | \( 1 + (-0.432 - 2.17i)T + (-17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (4.52 - 1.87i)T + (16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (3.43 - 5.14i)T + (-11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 + 2.88iT - 31T^{2} \) |
| 37 | \( 1 + (-1.20 + 6.05i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (3.20 - 1.32i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (2.16 - 1.44i)T + (16.4 - 39.7i)T^{2} \) |
| 47 | \( 1 + (2.37 - 2.37i)T - 47iT^{2} \) |
| 53 | \( 1 + (3.20 + 4.78i)T + (-20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (-1.01 - 0.201i)T + (54.5 + 22.5i)T^{2} \) |
| 61 | \( 1 + (2.23 + 1.49i)T + (23.3 + 56.3i)T^{2} \) |
| 67 | \( 1 + (-12.1 - 8.10i)T + (25.6 + 61.8i)T^{2} \) |
| 71 | \( 1 + (-4.63 + 11.1i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-3.99 - 9.65i)T + (-51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (-1.05 - 1.05i)T + 79iT^{2} \) |
| 83 | \( 1 + (1.67 + 8.41i)T + (-76.6 + 31.7i)T^{2} \) |
| 89 | \( 1 + (5.40 + 2.23i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + 11.0iT - 97T^{2} \) |
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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
−14.69569856424848089101143624405, −14.12220731630810834959693172038, −12.51789559960558111385914775597, −11.30782780020643035935096512808, −10.07218871838813822039136534082, −9.467510238772946446548616490219, −7.43955301352205010031802805687, −6.59016377013267736394643063991, −5.42285809935924254823412373823, −1.75938403609742759633267192590,
2.26220342849428405644440216779, 5.25741884765450467628769251730, 6.25935828322431037464875362895, 8.310866851320026859516210325379, 9.491818124075770967416107871797, 10.22705080668161986654693958252, 11.43603749504767646758271143342, 12.39235774501615566528257434438, 13.83498518052625347968909266147, 15.25549136197156217268070214024
