| L(s) = 1 | + (1.21 − 0.783i)3-s + (−1.24 − 2.72i)5-s + (1.62 − 0.476i)7-s + (−0.374 + 0.819i)9-s + (−3.72 − 4.30i)11-s + (6.20 + 1.82i)13-s + (−3.65 − 2.34i)15-s + (−0.846 + 5.88i)17-s + (−0.0398 − 0.277i)19-s + (1.60 − 1.85i)21-s + (4.79 + 0.152i)23-s + (−2.60 + 3.00i)25-s + (0.804 + 5.59i)27-s + (−0.171 + 1.19i)29-s + (−2.51 − 1.61i)31-s + ⋯ |
| L(s) = 1 | + (0.703 − 0.452i)3-s + (−0.556 − 1.21i)5-s + (0.613 − 0.180i)7-s + (−0.124 + 0.273i)9-s + (−1.12 − 1.29i)11-s + (1.72 + 0.505i)13-s + (−0.943 − 0.606i)15-s + (−0.205 + 1.42i)17-s + (−0.00914 − 0.0636i)19-s + (0.350 − 0.404i)21-s + (0.999 + 0.0318i)23-s + (−0.521 + 0.601i)25-s + (0.154 + 1.07i)27-s + (−0.0318 + 0.221i)29-s + (−0.452 − 0.290i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 + 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.17780 - 0.676887i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17780 - 0.676887i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (-4.79 - 0.152i)T \) |
| good | 3 | \( 1 + (-1.21 + 0.783i)T + (1.24 - 2.72i)T^{2} \) |
| 5 | \( 1 + (1.24 + 2.72i)T + (-3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-1.62 + 0.476i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (3.72 + 4.30i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-6.20 - 1.82i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.846 - 5.88i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (0.0398 + 0.277i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (0.171 - 1.19i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (2.51 + 1.61i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-0.110 + 0.241i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-2.41 - 5.28i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (4.76 - 3.06i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 1.51T + 47T^{2} \) |
| 53 | \( 1 + (6.33 - 1.86i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (3.08 + 0.904i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-7.37 - 4.73i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-2.03 + 2.35i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-1.43 + 1.65i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (1.80 + 12.5i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (0.627 + 0.184i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-5.74 + 12.5i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (8.24 - 5.30i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-1.58 - 3.48i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−12.92811843275051599363663195083, −11.28899293474812526300103826647, −10.83813804202731275956170555834, −8.853607496407598308558811017223, −8.405270895449401741489725151336, −7.82567365961921959247679348088, −6.03429502214893981657718624035, −4.78803942826834429907070512522, −3.40680791662783373117589691476, −1.43105618184425596046208302133,
2.63465183618293417954930624899, 3.65257466543641109055315674648, 5.11751790548321431867806998235, 6.76227015939532685263250387290, 7.71969588610836089703328577275, 8.693162825883491896096965854519, 9.865024514413968525327044125087, 10.85689388687598330458403046449, 11.51985404888590515308120669760, 12.87292116087706477508592766966
