| L(s) = 1 | + (−1.17 − 0.784i)2-s + (−1.47 − 0.907i)3-s + (0.769 + 1.84i)4-s + (−2.36 − 2.70i)5-s + (1.02 + 2.22i)6-s + (−4.85 − 0.639i)7-s + (0.543 − 2.77i)8-s + (1.35 + 2.67i)9-s + (0.668 + 5.03i)10-s + (−2.39 − 4.85i)11-s + (0.540 − 3.42i)12-s + (−1.38 − 4.08i)13-s + (5.21 + 4.56i)14-s + (1.04 + 6.13i)15-s + (−2.81 + 2.83i)16-s + (2.53 + 2.53i)17-s + ⋯ |
| L(s) = 1 | + (−0.832 − 0.554i)2-s + (−0.851 − 0.523i)3-s + (0.384 + 0.923i)4-s + (−1.05 − 1.20i)5-s + (0.418 + 0.908i)6-s + (−1.83 − 0.241i)7-s + (0.192 − 0.981i)8-s + (0.451 + 0.892i)9-s + (0.211 + 1.59i)10-s + (−0.721 − 1.46i)11-s + (0.155 − 0.987i)12-s + (−0.384 − 1.13i)13-s + (1.39 + 1.21i)14-s + (0.269 + 1.58i)15-s + (−0.704 + 0.709i)16-s + (0.614 + 0.614i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0816674 + 0.0372548i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0816674 + 0.0372548i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.17 + 0.784i)T \) |
| 3 | \( 1 + (1.47 + 0.907i)T \) |
| good | 5 | \( 1 + (2.36 + 2.70i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (4.85 + 0.639i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (2.39 + 4.85i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (1.38 + 4.08i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (-2.53 - 2.53i)T + 17iT^{2} \) |
| 19 | \( 1 + (0.255 - 1.28i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (0.535 + 4.07i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (1.21 - 0.0796i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (-0.338 + 0.195i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.208 + 1.04i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (0.197 + 1.49i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (1.18 - 0.584i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (-1.28 + 0.343i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-1.84 + 2.76i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (-6.59 - 7.52i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (-0.245 - 3.74i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (2.51 + 1.24i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (5.41 + 13.0i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-5.64 + 13.6i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-1.16 - 4.36i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (2.73 + 2.39i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (2.03 - 0.844i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (-8.16 - 4.71i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.28077065928658762450137704289, −9.035386370177283448831128770709, −8.140750845145778391922271970214, −7.60084415522473757233131188775, −6.39152375904435159084911382506, −5.46681940124766461666034692367, −3.92941049455155890390673970780, −2.98175947663301533442951299903, −0.74695007252706403223511185805, −0.11648341180153700443641835534,
2.67791148276970967902273619810, 3.94610286095852758506981842278, 5.24390871271413602428949744552, 6.46557155251358609053768561872, 6.98858461070927678396747329075, 7.51421189841016391365772885110, 9.192294355478646343028302441905, 9.903573305853514553316997889423, 10.22814591094470385057332848009, 11.37990238558602044639238522844
