| L(s) = 1 | + (−0.0998 + 1.33i)2-s + (0.211 + 0.0319i)4-s + (−2.75 − 2.56i)5-s + (0.0649 + 2.64i)7-s + (−0.658 + 2.88i)8-s + (3.68 − 3.42i)10-s + (3.72 + 2.53i)11-s + (−0.0589 + 0.0283i)13-s + (−3.53 − 0.177i)14-s + (−3.36 − 1.03i)16-s + (−2.49 + 6.35i)17-s + (−1.99 + 3.45i)19-s + (−0.503 − 0.630i)20-s + (−3.75 + 4.70i)22-s + (−1.55 − 3.95i)23-s + ⋯ |
| L(s) = 1 | + (−0.0706 + 0.942i)2-s + (0.105 + 0.0159i)4-s + (−1.23 − 1.14i)5-s + (0.0245 + 0.999i)7-s + (−0.232 + 1.01i)8-s + (1.16 − 1.08i)10-s + (1.12 + 0.765i)11-s + (−0.0163 + 0.00787i)13-s + (−0.943 − 0.0474i)14-s + (−0.842 − 0.259i)16-s + (−0.605 + 1.54i)17-s + (−0.457 + 0.792i)19-s + (−0.112 − 0.141i)20-s + (−0.800 + 1.00i)22-s + (−0.323 − 0.824i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.399157 + 0.994983i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.399157 + 0.994983i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.0649 - 2.64i)T \) |
| good | 2 | \( 1 + (0.0998 - 1.33i)T + (-1.97 - 0.298i)T^{2} \) |
| 5 | \( 1 + (2.75 + 2.56i)T + (0.373 + 4.98i)T^{2} \) |
| 11 | \( 1 + (-3.72 - 2.53i)T + (4.01 + 10.2i)T^{2} \) |
| 13 | \( 1 + (0.0589 - 0.0283i)T + (8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (2.49 - 6.35i)T + (-12.4 - 11.5i)T^{2} \) |
| 19 | \( 1 + (1.99 - 3.45i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.55 + 3.95i)T + (-16.8 + 15.6i)T^{2} \) |
| 29 | \( 1 + (-5.59 - 7.02i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (2.05 + 3.55i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.91 + 0.288i)T + (35.3 - 10.9i)T^{2} \) |
| 41 | \( 1 + (-1.34 + 5.88i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (0.175 + 0.770i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (0.0438 - 0.585i)T + (-46.4 - 7.00i)T^{2} \) |
| 53 | \( 1 + (-3.00 - 0.452i)T + (50.6 + 15.6i)T^{2} \) |
| 59 | \( 1 + (-1.36 + 1.27i)T + (4.40 - 58.8i)T^{2} \) |
| 61 | \( 1 + (-4.88 + 0.735i)T + (58.2 - 17.9i)T^{2} \) |
| 67 | \( 1 + (1.53 + 2.65i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.78 + 4.74i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-0.783 - 10.4i)T + (-72.1 + 10.8i)T^{2} \) |
| 79 | \( 1 + (-6.49 + 11.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.60 + 1.25i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (-9.50 + 6.48i)T + (32.5 - 82.8i)T^{2} \) |
| 97 | \( 1 - 9.43T + 97T^{2} \) |
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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
−11.77820914194257232674170473770, −10.67436181198003325689213349086, −9.102623976759939782516655972005, −8.548362164230571083246648610155, −7.969465455396667100821593405791, −6.80836927691918883312082339123, −5.96054991537318725533935583790, −4.80177411568067110877128603465, −3.88261155249398853159406229477, −1.90864828772534107486048554036,
0.69812101542231532706438932000, 2.69578470240148259850041063800, 3.59034846338220927120483558781, 4.36491773814548981572366414888, 6.54089170697738687219156891423, 6.94175430201640886683026065061, 7.904705231512276702731953696129, 9.251943863924464812110500439117, 10.20125307053422785324957637428, 11.11600674857620441243505128970
