| L(s) = 1 | + (−0.142 + 0.989i)3-s + (−0.911 + 0.267i)5-s + (−1.46 − 1.69i)7-s + (−0.959 − 0.281i)9-s + (−3.17 − 2.04i)11-s + (−0.129 + 0.149i)13-s + (−0.135 − 0.939i)15-s + (−1.19 − 2.62i)17-s + (1.90 − 4.16i)19-s + (1.88 − 1.20i)21-s + (−4.06 − 2.54i)23-s + (−3.44 + 2.21i)25-s + (0.415 − 0.909i)27-s + (0.572 + 1.25i)29-s + (−0.120 − 0.839i)31-s + ⋯ |
| L(s) = 1 | + (−0.0821 + 0.571i)3-s + (−0.407 + 0.119i)5-s + (−0.553 − 0.638i)7-s + (−0.319 − 0.0939i)9-s + (−0.957 − 0.615i)11-s + (−0.0358 + 0.0413i)13-s + (−0.0348 − 0.242i)15-s + (−0.290 − 0.636i)17-s + (0.436 − 0.955i)19-s + (0.410 − 0.263i)21-s + (−0.847 − 0.531i)23-s + (−0.689 + 0.443i)25-s + (0.0799 − 0.175i)27-s + (0.106 + 0.232i)29-s + (−0.0216 − 0.150i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.259397 - 0.416911i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.259397 - 0.416911i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.142 - 0.989i)T \) |
| 23 | \( 1 + (4.06 + 2.54i)T \) |
| good | 5 | \( 1 + (0.911 - 0.267i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (1.46 + 1.69i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (3.17 + 2.04i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (0.129 - 0.149i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.19 + 2.62i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.90 + 4.16i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.572 - 1.25i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (0.120 + 0.839i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-5.57 - 1.63i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-1.39 + 0.410i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-1.75 + 12.2i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 13.0T + 47T^{2} \) |
| 53 | \( 1 + (0.0705 + 0.0814i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (3.66 - 4.22i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.04 - 7.25i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (4.74 - 3.05i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (11.1 - 7.18i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-2.58 + 5.65i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (0.0200 - 0.0231i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-9.43 - 2.77i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.111 + 0.773i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (2.46 - 0.724i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.48127968332832027228649454079, −9.766951564848026002878181770928, −8.805216815264862938194105235253, −7.79587437850669827387144091548, −6.94881827984941600180088386286, −5.80757374554569070587047715127, −4.76445032250296137334972827469, −3.72037423053009759748664019766, −2.70957622071971782454340969911, −0.26953550587083667374558675190,
1.90174468465250271213583321634, 3.14425650983161879779859084693, 4.48756591548322206593943216104, 5.73521937068741563425289461914, 6.40965156187838165938610665084, 7.83049398980864355936459148101, 8.004961635881908117554750880723, 9.407478806421837698332981214552, 10.09101876278027981745308088005, 11.20644602510791191774699450768
