| L(s) = 1 | + (−2.76 − 4.78i)2-s + (−13.5 − 7.79i)3-s + (16.7 − 28.9i)4-s + (−57.9 + 33.4i)5-s + 86.1i·6-s + (−240. + 244. i)7-s − 538.·8-s + (121.5 + 210. i)9-s + (320. + 185. i)10-s + (−862. + 1.49e3i)11-s + (−451. + 260. i)12-s − 2.80e3i·13-s + (1.83e3 + 475. i)14-s + 1.04e3·15-s + (418. + 724. i)16-s + (−5.32e3 − 3.07e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.345 − 0.598i)2-s + (−0.5 − 0.288i)3-s + (0.261 − 0.452i)4-s + (−0.463 + 0.267i)5-s + 0.398i·6-s + (−0.701 + 0.713i)7-s − 1.05·8-s + (0.166 + 0.288i)9-s + (0.320 + 0.185i)10-s + (−0.648 + 1.12i)11-s + (−0.261 + 0.150i)12-s − 1.27i·13-s + (0.668 + 0.173i)14-s + 0.309·15-s + (0.102 + 0.176i)16-s + (−1.08 − 0.625i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.796 - 0.605i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.796 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0596260 + 0.176928i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0596260 + 0.176928i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (13.5 + 7.79i)T \) |
| 7 | \( 1 + (240. - 244. i)T \) |
| good | 2 | \( 1 + (2.76 + 4.78i)T + (-32 + 55.4i)T^{2} \) |
| 5 | \( 1 + (57.9 - 33.4i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 11 | \( 1 + (862. - 1.49e3i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + 2.80e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (5.32e3 + 3.07e3i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-7.73e3 + 4.46e3i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (4.95e3 + 8.57e3i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + 1.36e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + (2.18e4 + 1.26e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (1.13e4 + 1.96e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 - 3.78e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 7.36e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-1.20e5 + 6.95e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-8.25e3 + 1.43e4i)T + (-1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (6.19e4 + 3.57e4i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (2.28e5 - 1.31e5i)T + (2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (2.74e5 - 4.75e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 - 4.65e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (2.89e5 + 1.66e5i)T + (7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-2.05e5 - 3.55e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + 7.19e4iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (1.41e5 - 8.16e4i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 + 6.51e5iT - 8.32e11T^{2} \) |
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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
−15.78816168372341428031146160422, −15.19573955171648518520809656251, −12.99103754814306468963072033154, −11.88441672589192859068209451728, −10.68547472855786870154883502280, −9.419935737028294501204421675218, −7.23122827869277078580575861095, −5.54566399907551456002992985096, −2.57102049400270543897504552507, −0.13003306221920434091949669750,
3.76214936758483888524351914236, 6.15600318161202427310998481392, 7.58838686491116326353844473490, 9.142094730630209064625482761074, 10.96272149688307756491253366552, 12.20637771806521707076266332608, 13.74505750249019307226426246585, 15.78082405908236448334012650172, 16.23719957122484010686127046635, 17.19167346997020197826579234769
