| L(s) = 1 | + (−0.637 − 0.112i)2-s + (−1.11 + 1.32i)3-s + (−3.36 − 1.22i)4-s + (−6.57 + 2.39i)5-s + (0.859 − 0.720i)6-s + (−1.12 − 1.95i)7-s + (4.25 + 2.45i)8-s + (−0.520 − 2.95i)9-s + (4.46 − 0.787i)10-s + (−5.72 + 9.92i)11-s + (5.37 − 3.10i)12-s + (−4.27 − 5.09i)13-s + (0.499 + 1.37i)14-s + (4.14 − 11.3i)15-s + (8.53 + 7.16i)16-s + (−2.16 + 12.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.318 − 0.0562i)2-s + (−0.371 + 0.442i)3-s + (−0.841 − 0.306i)4-s + (−1.31 + 0.478i)5-s + (0.143 − 0.120i)6-s + (−0.161 − 0.279i)7-s + (0.531 + 0.306i)8-s + (−0.0578 − 0.328i)9-s + (0.446 − 0.0787i)10-s + (−0.520 + 0.902i)11-s + (0.447 − 0.258i)12-s + (−0.328 − 0.391i)13-s + (0.0356 + 0.0980i)14-s + (0.276 − 0.759i)15-s + (0.533 + 0.447i)16-s + (−0.127 + 0.720i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.214i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.976 - 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0163072 + 0.150083i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0163072 + 0.150083i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.11 - 1.32i)T \) |
| 19 | \( 1 + (-12.4 + 14.3i)T \) |
| good | 2 | \( 1 + (0.637 + 0.112i)T + (3.75 + 1.36i)T^{2} \) |
| 5 | \( 1 + (6.57 - 2.39i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (1.12 + 1.95i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (5.72 - 9.92i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (4.27 + 5.09i)T + (-29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (2.16 - 12.2i)T + (-271. - 98.8i)T^{2} \) |
| 23 | \( 1 + (20.3 + 7.39i)T + (405. + 340. i)T^{2} \) |
| 29 | \( 1 + (6.16 - 1.08i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (44.9 - 25.9i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 47.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-4.93 + 5.87i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (49.4 - 18.0i)T + (1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (12.5 + 71.2i)T + (-2.07e3 + 755. i)T^{2} \) |
| 53 | \( 1 + (24.2 - 66.6i)T + (-2.15e3 - 1.80e3i)T^{2} \) |
| 59 | \( 1 + (-66.4 - 11.7i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-108. - 39.3i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (64.2 - 11.3i)T + (4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (35.7 + 98.2i)T + (-3.86e3 + 3.24e3i)T^{2} \) |
| 73 | \( 1 + (-81.9 - 68.7i)T + (925. + 5.24e3i)T^{2} \) |
| 79 | \( 1 + (-2.37 + 2.83i)T + (-1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-33.9 - 58.7i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (64.2 + 76.5i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (-71.7 - 12.6i)T + (8.84e3 + 3.21e3i)T^{2} \) |
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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
−15.37076056828041516453661340849, −14.72157509903701813010068335933, −13.19224961714257139436816513198, −11.95845304460742542214489147424, −10.70032946397570588140532956228, −9.888983817678814576683815723734, −8.368970057995563221032144215687, −7.17260948961516196665160169528, −5.07878812859442351318682879531, −3.81755941047252285258935327534,
0.17038282502028985804219824388, 3.82201614099165696230455230440, 5.38076769320746089987827101633, 7.48871406088154292552945606889, 8.255996923128524807308796770317, 9.512871919991449573034354999813, 11.26094235513362989494468687656, 12.19650258552213210461936905254, 13.14352499511356555632416220960, 14.33588056796864062254211413241
