| L(s) = 1 | + (−0.474 − 0.930i)2-s + (−0.440 + 2.78i)3-s + (0.533 − 0.734i)4-s + (2.23 + 0.0540i)5-s + (2.79 − 0.909i)6-s + (−0.543 + 0.0860i)7-s + (−3.00 − 0.475i)8-s + (−4.68 − 1.52i)9-s + (−1.01 − 2.10i)10-s + (−3.29 − 0.335i)11-s + (1.80 + 1.80i)12-s + (2.89 − 1.47i)13-s + (0.337 + 0.464i)14-s + (−1.13 + 6.19i)15-s + (0.419 + 1.29i)16-s + (−5.04 − 2.57i)17-s + ⋯ |
| L(s) = 1 | + (−0.335 − 0.658i)2-s + (−0.254 + 1.60i)3-s + (0.266 − 0.367i)4-s + (0.999 + 0.0241i)5-s + (1.14 − 0.371i)6-s + (−0.205 + 0.0325i)7-s + (−1.06 − 0.168i)8-s + (−1.56 − 0.507i)9-s + (−0.319 − 0.666i)10-s + (−0.994 − 0.101i)11-s + (0.522 + 0.522i)12-s + (0.801 − 0.408i)13-s + (0.0902 + 0.124i)14-s + (−0.293 + 1.59i)15-s + (0.104 + 0.322i)16-s + (−1.22 − 0.623i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.769382 + 0.0353856i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.769382 + 0.0353856i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.23 - 0.0540i)T \) |
| 11 | \( 1 + (3.29 + 0.335i)T \) |
| good | 2 | \( 1 + (0.474 + 0.930i)T + (-1.17 + 1.61i)T^{2} \) |
| 3 | \( 1 + (0.440 - 2.78i)T + (-2.85 - 0.927i)T^{2} \) |
| 7 | \( 1 + (0.543 - 0.0860i)T + (6.65 - 2.16i)T^{2} \) |
| 13 | \( 1 + (-2.89 + 1.47i)T + (7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (5.04 + 2.57i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-1.25 + 0.914i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (0.803 - 0.803i)T - 23iT^{2} \) |
| 29 | \( 1 + (-3.44 - 2.50i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (0.509 - 1.56i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.149 + 0.945i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-5.25 - 7.23i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-2.55 - 2.55i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.02 - 0.636i)T + (44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (3.19 + 6.27i)T + (-31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-3.97 + 5.47i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (8.75 - 2.84i)T + (49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (2.62 + 2.62i)T + 67iT^{2} \) |
| 71 | \( 1 + (-2.11 - 6.51i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.57 - 9.96i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-1.28 + 3.96i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-5.14 + 10.0i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + 3.64iT - 89T^{2} \) |
| 97 | \( 1 + (-14.8 + 7.56i)T + (57.0 - 78.4i)T^{2} \) |
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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
−15.68791991476255340378338810510, −14.43294072849954603034576975303, −13.06871289874405598854395185784, −11.25440439795185372817309568595, −10.61413970828910872431206934616, −9.776804341246773447516730564630, −8.926292763870742986135841980245, −6.18452953024848744760122570952, −5.03398526570061603130591842774, −2.90621412093884840572391895095,
2.30201715349521540754721535038, 5.91964563116307206628612997750, 6.64131331392991508178295589584, 7.80962763772320751283150377086, 8.923872544738054384216099954414, 10.83980658820952966517355792890, 12.23867769931468976673391616349, 13.11349212285347358121412436226, 13.87455343802352467968095923407, 15.51754999881598381413867315642
