| L(s) = 1 | + (0.654 − 0.755i)3-s + (−0.280 + 1.95i)5-s + (1.41 + 3.10i)7-s + (−0.142 − 0.989i)9-s + (−5.15 + 1.51i)11-s + (−2.46 + 5.39i)13-s + (1.29 + 1.48i)15-s + (−0.305 + 0.196i)17-s + (−0.994 − 0.639i)19-s + (3.27 + 0.960i)21-s + (1.02 − 4.68i)23-s + (1.06 + 0.313i)25-s + (−0.841 − 0.540i)27-s + (0.983 − 0.632i)29-s + (−0.519 − 0.599i)31-s + ⋯ |
| L(s) = 1 | + (0.378 − 0.436i)3-s + (−0.125 + 0.872i)5-s + (0.535 + 1.17i)7-s + (−0.0474 − 0.329i)9-s + (−1.55 + 0.456i)11-s + (−0.683 + 1.49i)13-s + (0.333 + 0.384i)15-s + (−0.0741 + 0.0476i)17-s + (−0.228 − 0.146i)19-s + (0.713 + 0.209i)21-s + (0.214 − 0.976i)23-s + (0.213 + 0.0627i)25-s + (−0.161 − 0.104i)27-s + (0.182 − 0.117i)29-s + (−0.0932 − 0.107i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0505 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0505 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.961465 + 0.914059i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.961465 + 0.914059i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (-1.02 + 4.68i)T \) |
| good | 5 | \( 1 + (0.280 - 1.95i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.41 - 3.10i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (5.15 - 1.51i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (2.46 - 5.39i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (0.305 - 0.196i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (0.994 + 0.639i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-0.983 + 0.632i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (0.519 + 0.599i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-1.18 - 8.20i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.106 + 0.738i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-4.42 + 5.11i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 + (-1.43 - 3.14i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-1.36 + 2.98i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-6.91 - 7.98i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-10.9 - 3.22i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (9.31 + 2.73i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-0.510 - 0.328i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-6.00 + 13.1i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.783 - 5.44i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (7.37 - 8.51i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (1.24 - 8.62i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−11.01553543331802902792460891697, −10.17748942364776141989461572002, −9.091202456099880743651816195018, −8.340380103914574319042361516443, −7.36013736511712079249488670123, −6.67199372426398868150312777420, −5.45734976986572462213009829830, −4.42670642945420714408509167164, −2.66489702358862255158346046800, −2.24538409208361957353544755632,
0.71471313243020694069059830407, 2.65402586372693624654793515519, 3.89780398359714576846208098632, 4.98036451312093255064500102635, 5.53530420386450870026459691947, 7.43636597281965117575348486350, 7.85856246297658775936135601296, 8.677213749051401854089881413129, 9.829654793274434807364476851649, 10.57176839388161958115367731239
