| L(s) = 1 | + (1.50 − 0.852i)3-s + (−2.12 − 0.375i)5-s + (−2.50 − 2.98i)7-s + (1.54 − 2.57i)9-s + (−0.0357 − 0.202i)11-s + (−2.01 + 0.733i)13-s + (−3.52 + 1.24i)15-s + (−5.87 − 3.38i)17-s + (6.56 − 3.78i)19-s + (−6.31 − 2.36i)21-s + (3.99 + 3.34i)23-s + (−0.312 − 0.113i)25-s + (0.141 − 5.19i)27-s + (−0.709 + 1.94i)29-s + (1.02 − 1.22i)31-s + ⋯ |
| L(s) = 1 | + (0.870 − 0.492i)3-s + (−0.951 − 0.167i)5-s + (−0.945 − 1.12i)7-s + (0.515 − 0.856i)9-s + (−0.0107 − 0.0611i)11-s + (−0.558 + 0.203i)13-s + (−0.910 + 0.322i)15-s + (−1.42 − 0.822i)17-s + (1.50 − 0.868i)19-s + (−1.37 − 0.515i)21-s + (0.832 + 0.698i)23-s + (−0.0624 − 0.0227i)25-s + (0.0271 − 0.999i)27-s + (−0.131 + 0.361i)29-s + (0.184 − 0.219i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.472 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.604617 - 1.01071i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.604617 - 1.01071i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.50 + 0.852i)T \) |
| good | 5 | \( 1 + (2.12 + 0.375i)T + (4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (2.50 + 2.98i)T + (-1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.0357 + 0.202i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (2.01 - 0.733i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (5.87 + 3.38i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.56 + 3.78i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.99 - 3.34i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (0.709 - 1.94i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.02 + 1.22i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-3.90 + 6.75i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.12 - 11.3i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-6.15 + 1.08i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (3.44 - 2.89i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 6.89iT - 53T^{2} \) |
| 59 | \( 1 + (0.0937 - 0.531i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-0.695 + 0.583i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (1.77 + 4.88i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-5.04 + 8.74i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.46 - 2.53i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.25 + 3.45i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-10.0 - 3.66i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (9.79 - 5.65i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.67 - 9.50i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.01309937838950036077769277778, −9.516254086867175039833753805791, −9.290625467713929329683249082777, −7.82892244112679922922122567443, −7.29200521351708956749933327943, −6.61087034123536153124438992674, −4.72038810409430127468474277957, −3.72529980652660241750292342268, −2.76731037385253440492503352136, −0.67082825356755548527873823040,
2.44608238701621822860160172766, 3.37118577265770369989972337896, 4.40725356351043721985460089545, 5.70164605289445777214769997275, 6.98629010117843932904646476116, 7.934686257161580422618784922331, 8.801545436128546302801800844608, 9.498657582905929642620713891975, 10.40864507741803068083934965425, 11.46443363253342904391220819131
