| L(s) = 1 | + (−1.20 − 0.740i)2-s + (1.89 + 1.29i)3-s + (0.903 + 1.78i)4-s + (0.459 + 0.0344i)5-s + (−1.32 − 2.96i)6-s + (−0.101 − 2.64i)7-s + (0.232 − 2.81i)8-s + (0.830 + 2.11i)9-s + (−0.527 − 0.381i)10-s + (5.59 + 2.19i)11-s + (−0.593 + 4.55i)12-s + (−4.33 + 3.45i)13-s + (−1.83 + 3.26i)14-s + (0.826 + 0.659i)15-s + (−2.36 + 3.22i)16-s + (2.11 + 2.27i)17-s + ⋯ |
| L(s) = 1 | + (−0.851 − 0.523i)2-s + (1.09 + 0.746i)3-s + (0.451 + 0.892i)4-s + (0.205 + 0.0153i)5-s + (−0.542 − 1.20i)6-s + (−0.0385 − 0.999i)7-s + (0.0821 − 0.996i)8-s + (0.276 + 0.705i)9-s + (−0.166 − 0.120i)10-s + (1.68 + 0.661i)11-s + (−0.171 + 1.31i)12-s + (−1.20 + 0.958i)13-s + (−0.490 + 0.871i)14-s + (0.213 + 0.170i)15-s + (−0.591 + 0.806i)16-s + (0.512 + 0.552i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0756i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.18347 + 0.0448257i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18347 + 0.0448257i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.20 + 0.740i)T \) |
| 7 | \( 1 + (0.101 + 2.64i)T \) |
| good | 3 | \( 1 + (-1.89 - 1.29i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (-0.459 - 0.0344i)T + (4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (-5.59 - 2.19i)T + (8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (4.33 - 3.45i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-2.11 - 2.27i)T + (-1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-3.06 + 5.30i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.35 - 3.61i)T + (-1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.395 - 1.73i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (1.92 + 3.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.78 + 0.859i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (4.16 + 8.64i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (0.134 - 0.278i)T + (-26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (3.46 - 0.522i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (0.191 - 0.0589i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (0.0565 + 0.755i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-0.954 + 3.09i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (10.0 - 5.77i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-10.3 - 2.35i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (0.0295 - 0.196i)T + (-69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (3.05 + 1.76i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.10 + 3.88i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (8.95 - 3.51i)T + (65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 - 4.04iT - 97T^{2} \) |
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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
−12.22040228669796435481207070605, −11.44185404150612083894896066002, −10.02141693609150239042730205875, −9.622819373740210976411086833874, −8.935814088493771064912686818900, −7.59578043160848638572520531422, −6.81906697059698325916702602444, −4.31191584899083156783835732289, −3.56508430542302785299325312111, −1.90043066135611398903724068388,
1.68117258163765906746113497289, 3.05058467500509944709768894448, 5.46656443866248143095391151958, 6.50491942071919870466823606178, 7.71364670242747805427893033139, 8.359887645651538499053196632029, 9.310738765825932583724860114749, 9.992284916154856698375899305315, 11.69958081233375998611709225529, 12.34051575998963503428534074606
