| L(s) = 1 | + (0.161 + 1.01i)2-s + (−1.71 − 0.245i)3-s + (0.888 − 0.288i)4-s + (2.26 + 1.15i)5-s + (−0.0270 − 1.78i)6-s + (−1.08 + 0.260i)7-s + (1.37 + 2.69i)8-s + (2.87 + 0.840i)9-s + (−0.812 + 2.49i)10-s + (−1.53 + 1.80i)11-s + (−1.59 + 0.277i)12-s + (−0.726 − 1.18i)13-s + (−0.441 − 1.06i)14-s + (−3.60 − 2.53i)15-s + (−1.01 + 0.739i)16-s + (1.52 − 0.120i)17-s + ⋯ |
| L(s) = 1 | + (0.114 + 0.720i)2-s + (−0.989 − 0.141i)3-s + (0.444 − 0.144i)4-s + (1.01 + 0.516i)5-s + (−0.0110 − 0.729i)6-s + (−0.410 + 0.0985i)7-s + (0.486 + 0.954i)8-s + (0.959 + 0.280i)9-s + (−0.256 + 0.790i)10-s + (−0.463 + 0.542i)11-s + (−0.460 + 0.0800i)12-s + (−0.201 − 0.328i)13-s + (−0.117 − 0.284i)14-s + (−0.931 − 0.655i)15-s + (−0.254 + 0.184i)16-s + (0.370 − 0.0291i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.490 - 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.490 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.935042 + 0.546401i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.935042 + 0.546401i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.71 + 0.245i)T \) |
| 41 | \( 1 + (-4.94 + 4.06i)T \) |
| good | 2 | \( 1 + (-0.161 - 1.01i)T + (-1.90 + 0.618i)T^{2} \) |
| 5 | \( 1 + (-2.26 - 1.15i)T + (2.93 + 4.04i)T^{2} \) |
| 7 | \( 1 + (1.08 - 0.260i)T + (6.23 - 3.17i)T^{2} \) |
| 11 | \( 1 + (1.53 - 1.80i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.726 + 1.18i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (-1.52 + 0.120i)T + (16.7 - 2.65i)T^{2} \) |
| 19 | \( 1 + (-3.34 + 5.45i)T + (-8.62 - 16.9i)T^{2} \) |
| 23 | \( 1 + (7.14 + 5.19i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (6.38 + 0.502i)T + (28.6 + 4.53i)T^{2} \) |
| 31 | \( 1 + (-5.33 - 1.73i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.788 + 2.42i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (6.72 - 1.06i)T + (40.8 - 13.2i)T^{2} \) |
| 47 | \( 1 + (1.86 - 7.77i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (-0.400 + 5.09i)T + (-52.3 - 8.29i)T^{2} \) |
| 59 | \( 1 + (-3.09 + 4.25i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.173 - 1.09i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-7.65 - 8.96i)T + (-10.4 + 66.1i)T^{2} \) |
| 71 | \( 1 + (0.729 + 0.622i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (0.605 - 0.605i)T - 73iT^{2} \) |
| 79 | \( 1 + (-5.43 + 13.1i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + 0.110iT - 83T^{2} \) |
| 89 | \( 1 + (-2.24 - 9.35i)T + (-79.2 + 40.4i)T^{2} \) |
| 97 | \( 1 + (3.51 - 2.99i)T + (15.1 - 95.8i)T^{2} \) |
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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
−13.68058436377458217424502493733, −12.62918235791112424569387530945, −11.48520702287431531973058495463, −10.43769773737340974171844364251, −9.752551126883131604780211845280, −7.73731381872008006066195941155, −6.71803669985705270532024975923, −5.97279064638702111838764697590, −4.99873821698246335787182639464, −2.31164622159160387923329930006,
1.65579621818252117936298618283, 3.68185772394314659222670339212, 5.43199441468815901540135139535, 6.27963562690155086019785913342, 7.71298121699070998608699083902, 9.792171130990033499252864763758, 10.01434065635119118722299075015, 11.36237874245772204883307622321, 12.08887985078454018558739493985, 13.02651384365203543491408273319
