| L(s) = 1 | + (2.78 + 1.11i)2-s + (1.72 + 0.187i)3-s + (3.63 + 3.44i)4-s + (−3.64 + 4.29i)5-s + (4.59 + 2.43i)6-s + (3.88 + 2.33i)7-s + (1.27 + 2.75i)8-s + (2.92 + 0.644i)9-s + (−14.9 + 7.92i)10-s + (6.59 + 0.357i)11-s + (5.62 + 6.61i)12-s + (−1.02 − 4.65i)13-s + (8.22 + 10.8i)14-s + (−7.08 + 6.70i)15-s + (−0.591 − 10.9i)16-s + (−12.2 + 7.35i)17-s + ⋯ |
| L(s) = 1 | + (1.39 + 0.555i)2-s + (0.573 + 0.0624i)3-s + (0.909 + 0.861i)4-s + (−0.729 + 0.858i)5-s + (0.765 + 0.405i)6-s + (0.554 + 0.333i)7-s + (0.159 + 0.344i)8-s + (0.325 + 0.0716i)9-s + (−1.49 + 0.792i)10-s + (0.599 + 0.0325i)11-s + (0.468 + 0.551i)12-s + (−0.0788 − 0.358i)13-s + (0.587 + 0.773i)14-s + (−0.472 + 0.447i)15-s + (−0.0369 − 0.681i)16-s + (−0.719 + 0.432i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.393 - 0.919i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.77698 + 1.83123i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.77698 + 1.83123i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.72 - 0.187i)T \) |
| 59 | \( 1 + (-57.7 - 12.0i)T \) |
| good | 2 | \( 1 + (-2.78 - 1.11i)T + (2.90 + 2.75i)T^{2} \) |
| 5 | \( 1 + (3.64 - 4.29i)T + (-4.04 - 24.6i)T^{2} \) |
| 7 | \( 1 + (-3.88 - 2.33i)T + (22.9 + 43.2i)T^{2} \) |
| 11 | \( 1 + (-6.59 - 0.357i)T + (120. + 13.0i)T^{2} \) |
| 13 | \( 1 + (1.02 + 4.65i)T + (-153. + 70.9i)T^{2} \) |
| 17 | \( 1 + (12.2 - 7.35i)T + (135. - 255. i)T^{2} \) |
| 19 | \( 1 + (-5.79 + 20.8i)T + (-309. - 186. i)T^{2} \) |
| 23 | \( 1 + (-7.28 + 4.93i)T + (195. - 491. i)T^{2} \) |
| 29 | \( 1 + (19.2 + 48.3i)T + (-610. + 578. i)T^{2} \) |
| 31 | \( 1 + (-6.84 + 1.90i)T + (823. - 495. i)T^{2} \) |
| 37 | \( 1 + (1.82 - 3.95i)T + (-886. - 1.04e3i)T^{2} \) |
| 41 | \( 1 + (44.2 - 65.3i)T + (-622. - 1.56e3i)T^{2} \) |
| 43 | \( 1 + (-23.0 + 1.25i)T + (1.83e3 - 199. i)T^{2} \) |
| 47 | \( 1 + (17.6 - 14.9i)T + (357. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + (22.5 - 42.6i)T + (-1.57e3 - 2.32e3i)T^{2} \) |
| 61 | \( 1 + (17.7 + 7.08i)T + (2.70e3 + 2.55e3i)T^{2} \) |
| 67 | \( 1 + (24.0 + 51.8i)T + (-2.90e3 + 3.42e3i)T^{2} \) |
| 71 | \( 1 + (-25.8 - 30.4i)T + (-815. + 4.97e3i)T^{2} \) |
| 73 | \( 1 + (-34.3 - 45.1i)T + (-1.42e3 + 5.13e3i)T^{2} \) |
| 79 | \( 1 + (-0.508 + 0.0552i)T + (6.09e3 - 1.34e3i)T^{2} \) |
| 83 | \( 1 + (-24.1 - 71.5i)T + (-5.48e3 + 4.16e3i)T^{2} \) |
| 89 | \( 1 + (47.6 - 18.9i)T + (5.75e3 - 5.44e3i)T^{2} \) |
| 97 | \( 1 + (-22.7 + 29.8i)T + (-2.51e3 - 9.06e3i)T^{2} \) |
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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
−12.96700754806571995168696045891, −11.75820702616434870107351996663, −11.13155311043952102869789925864, −9.569914187583060530731834350905, −8.210334932774400155695570708660, −7.20499176483717686503169867222, −6.31175582802366064764863569291, −4.84579536088330384494631817812, −3.84588405191860310045965260020, −2.70634292345339978330731918840,
1.68551940126968885505140177906, 3.49053465142808918191432222164, 4.33691548449043542256866126366, 5.28223155180372679686092722423, 6.94063803956327375742334888723, 8.233786919103321624243425994514, 9.130284762887251106174477142701, 10.71729436226572249680142075332, 11.71728599036537626227347618209, 12.32567786274611215594039478462
