| L(s) = 1 | + (0.739 − 0.0706i)2-s + (0.690 + 0.723i)3-s + (−1.42 + 0.274i)4-s + (−3.42 − 1.76i)5-s + (0.561 + 0.486i)6-s + (2.62 − 0.344i)7-s + (−2.45 + 0.721i)8-s + (−0.0475 + 0.998i)9-s + (−2.65 − 1.06i)10-s + (0.416 − 4.36i)11-s + (−1.17 − 0.839i)12-s + (−5.31 + 0.763i)13-s + (1.91 − 0.440i)14-s + (−1.08 − 3.69i)15-s + (0.921 − 0.368i)16-s + (−1.85 − 5.34i)17-s + ⋯ |
| L(s) = 1 | + (0.523 − 0.0499i)2-s + (0.398 + 0.417i)3-s + (−0.710 + 0.137i)4-s + (−1.53 − 0.789i)5-s + (0.229 + 0.198i)6-s + (0.991 − 0.130i)7-s + (−0.869 + 0.255i)8-s + (−0.0158 + 0.332i)9-s + (−0.840 − 0.336i)10-s + (0.125 − 1.31i)11-s + (−0.340 − 0.242i)12-s + (−1.47 + 0.211i)13-s + (0.512 − 0.117i)14-s + (−0.280 − 0.954i)15-s + (0.230 − 0.0922i)16-s + (−0.448 − 1.29i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.582 + 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.318584 - 0.620231i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.318584 - 0.620231i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.690 - 0.723i)T \) |
| 7 | \( 1 + (-2.62 + 0.344i)T \) |
| 23 | \( 1 + (3.27 + 3.50i)T \) |
| good | 2 | \( 1 + (-0.739 + 0.0706i)T + (1.96 - 0.378i)T^{2} \) |
| 5 | \( 1 + (3.42 + 1.76i)T + (2.90 + 4.07i)T^{2} \) |
| 11 | \( 1 + (-0.416 + 4.36i)T + (-10.8 - 2.08i)T^{2} \) |
| 13 | \( 1 + (5.31 - 0.763i)T + (12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (1.85 + 5.34i)T + (-13.3 + 10.5i)T^{2} \) |
| 19 | \( 1 + (-0.381 + 1.10i)T + (-14.9 - 11.7i)T^{2} \) |
| 29 | \( 1 + (1.18 - 1.36i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (7.88 + 1.91i)T + (27.5 + 14.2i)T^{2} \) |
| 37 | \( 1 + (0.105 + 0.00500i)T + (36.8 + 3.51i)T^{2} \) |
| 41 | \( 1 + (-5.99 - 9.32i)T + (-17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (0.630 - 2.14i)T + (-36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (-4.42 + 2.55i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.31 - 1.66i)T + (-12.4 - 51.5i)T^{2} \) |
| 59 | \( 1 + (2.87 - 7.17i)T + (-42.7 - 40.7i)T^{2} \) |
| 61 | \( 1 + (-2.23 - 2.13i)T + (2.90 + 60.9i)T^{2} \) |
| 67 | \( 1 + (-7.65 + 5.44i)T + (21.9 - 63.3i)T^{2} \) |
| 71 | \( 1 + (-0.511 + 1.11i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (1.37 + 7.12i)T + (-67.7 + 27.1i)T^{2} \) |
| 79 | \( 1 + (-2.80 - 3.56i)T + (-18.6 + 76.7i)T^{2} \) |
| 83 | \( 1 + (11.1 + 7.14i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (1.96 + 8.10i)T + (-79.1 + 40.7i)T^{2} \) |
| 97 | \( 1 + (11.3 - 7.32i)T + (40.2 - 88.2i)T^{2} \) |
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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
−11.01927612894907477668472642531, −9.462813729254626315422282736471, −8.800213896801990971618977655703, −8.076468860090374972780732465758, −7.37525266303417289242162675836, −5.41416709232253211812004744897, −4.62813215588180696807691892920, −4.11922710672204394947530858406, −2.91469619072946367669041091115, −0.34150636991960680926445345002,
2.18033329540400681678694216085, 3.75245506354781029966077897778, 4.32836652365651129284457076636, 5.45531607363961559405721917657, 7.00416772671256215487811477215, 7.62530238653240198091466739644, 8.331345091883251159628987081524, 9.457881150439714769153000176852, 10.49362409689311781795127161682, 11.51225497786658089713243525284
