| L(s) = 1 | + (−0.309 − 0.951i)2-s + (−1.72 + 0.175i)3-s + (−0.809 + 0.587i)4-s + (−2.22 + 0.183i)5-s + (0.699 + 1.58i)6-s + (0.318 − 0.715i)7-s + (0.809 + 0.587i)8-s + (2.93 − 0.606i)9-s + (0.862 + 2.06i)10-s + (−0.402 − 3.82i)11-s + (1.29 − 1.15i)12-s + (−1.66 − 1.85i)13-s + (−0.778 − 0.0818i)14-s + (3.80 − 0.707i)15-s + (0.309 − 0.951i)16-s + (−3.59 − 0.377i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.994 + 0.101i)3-s + (−0.404 + 0.293i)4-s + (−0.996 + 0.0818i)5-s + (0.285 + 0.646i)6-s + (0.120 − 0.270i)7-s + (0.286 + 0.207i)8-s + (0.979 − 0.202i)9-s + (0.272 + 0.652i)10-s + (−0.121 − 1.15i)11-s + (0.372 − 0.333i)12-s + (−0.462 − 0.514i)13-s + (−0.208 − 0.0218i)14-s + (0.983 − 0.182i)15-s + (0.0772 − 0.237i)16-s + (−0.872 − 0.0916i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 - 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.215057 + 0.141972i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.215057 + 0.141972i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 3 | \( 1 + (1.72 - 0.175i)T \) |
| 5 | \( 1 + (2.22 - 0.183i)T \) |
| 31 | \( 1 + (-4.10 - 3.76i)T \) |
| good | 7 | \( 1 + (-0.318 + 0.715i)T + (-4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (0.402 + 3.82i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (1.66 + 1.85i)T + (-1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (3.59 + 0.377i)T + (16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (-1.23 + 1.37i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (4.03 - 5.55i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (1.50 + 4.63i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-2.63 - 4.56i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.59 - 7.52i)T + (-37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (5.98 - 6.64i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (-2.79 + 8.59i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-3.31 - 7.44i)T + (-35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (0.995 + 4.68i)T + (-53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 - 4.37iT - 61T^{2} \) |
| 67 | \( 1 + (-1.21 - 0.698i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.80 - 6.29i)T + (-47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (-0.874 - 8.31i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (14.5 + 1.52i)T + (77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-1.54 + 7.27i)T + (-75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (9.99 - 7.26i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (1.01 + 1.39i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−10.37828553785193823511504869992, −9.690602026551269766697375368602, −8.500288175175611862322974485871, −7.79411674790546433096732011564, −6.88385493989643297603413210161, −5.79787537232229903264861313084, −4.76845641983802044888723130628, −3.96238956338286836531063111854, −2.90951884881198306427335572803, −1.03447078453941393491392447609,
0.18555382941518327530864817134, 2.03182021615568501460682836462, 4.10625690443838602725662956233, 4.63589217965647836856186472919, 5.58182873107560657750642627827, 6.73247968718535923760322005259, 7.17480539871979082995584189932, 8.048926727982153301021827074933, 8.982322696104190117090610688567, 9.964863206165255129338927310601
