| 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
−11.37990238558602044639238522844, −10.22814591094470385057332848009, −9.903573305853514553316997889423, −9.192294355478646343028302441905, −7.51421189841016391365772885110, −6.98858461070927678396747329075, −6.46557155251358609053768561872, −5.24390871271413602428949744552, −3.94610286095852758506981842278, −2.67791148276970967902273619810,
0.11648341180153700443641835534, 0.74695007252706403223511185805, 2.98175947663301533442951299903, 3.92941049455155890390673970780, 5.46681940124766461666034692367, 6.39152375904435159084911382506, 7.60084415522473757233131188775, 8.140750845145778391922271970214, 9.035386370177283448831128770709, 10.28077065928658762450137704289
