| L(s) = 1 | + (0.118 + 1.57i)2-s + (−0.494 + 0.0745i)4-s + (−0.291 + 0.270i)5-s + (2.43 − 1.02i)7-s + (0.527 + 2.31i)8-s + (−0.460 − 0.427i)10-s + (1.83 − 1.25i)11-s + (1.92 + 0.928i)13-s + (1.91 + 3.72i)14-s + (−4.53 + 1.40i)16-s + (−0.243 − 0.620i)17-s + (0.781 + 1.35i)19-s + (0.123 − 0.155i)20-s + (2.18 + 2.74i)22-s + (0.231 − 0.589i)23-s + ⋯ |
| L(s) = 1 | + (0.0835 + 1.11i)2-s + (−0.247 + 0.0372i)4-s + (−0.130 + 0.120i)5-s + (0.921 − 0.389i)7-s + (0.186 + 0.817i)8-s + (−0.145 − 0.135i)10-s + (0.552 − 0.376i)11-s + (0.534 + 0.257i)13-s + (0.510 + 0.994i)14-s + (−1.13 + 0.350i)16-s + (−0.0591 − 0.150i)17-s + (0.179 + 0.310i)19-s + (0.0276 − 0.0347i)20-s + (0.466 + 0.584i)22-s + (0.0482 − 0.122i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0408 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0408 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20649 + 1.25683i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20649 + 1.25683i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.43 + 1.02i)T \) |
| good | 2 | \( 1 + (-0.118 - 1.57i)T + (-1.97 + 0.298i)T^{2} \) |
| 5 | \( 1 + (0.291 - 0.270i)T + (0.373 - 4.98i)T^{2} \) |
| 11 | \( 1 + (-1.83 + 1.25i)T + (4.01 - 10.2i)T^{2} \) |
| 13 | \( 1 + (-1.92 - 0.928i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (0.243 + 0.620i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (-0.781 - 1.35i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.231 + 0.589i)T + (-16.8 - 15.6i)T^{2} \) |
| 29 | \( 1 + (3.83 - 4.80i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (-1.69 + 2.93i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.20 + 0.633i)T + (35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (1.37 + 6.02i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (1.37 - 6.03i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (0.409 + 5.46i)T + (-46.4 + 7.00i)T^{2} \) |
| 53 | \( 1 + (-9.05 + 1.36i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (6.10 + 5.66i)T + (4.40 + 58.8i)T^{2} \) |
| 61 | \( 1 + (9.38 + 1.41i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (-0.892 + 1.54i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.30 - 1.63i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-1.08 + 14.4i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (-0.821 - 1.42i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.20 + 2.50i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-10.5 - 7.21i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + 9.59T + 97T^{2} \) |
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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
−11.24926452394508002109216547066, −10.68882917288761720033006922109, −9.207940720954921437416814243205, −8.387977182593084140096165362534, −7.54405437882810109738667885265, −6.80429955459498371622437333784, −5.77888496901080902028325489111, −4.87081485451623813870891305394, −3.62967182660816867561032649871, −1.72333503572812696125501856894,
1.32096277860122031846973984058, 2.48515941534882716903292848345, 3.81744913625635441274529066882, 4.75683179585845580450610229842, 6.10598343407540356377090687622, 7.26608676588492998145567240474, 8.338448690508616605732932171751, 9.253134626455618900810246500452, 10.25678471408641280440226578375, 11.01802289997036051920986274417
