| L(s) = 1 | + (−1.41 − 0.0214i)2-s + (−0.417 + 2.90i)3-s + (1.99 + 0.0606i)4-s + (−0.959 + 0.281i)5-s + (0.652 − 4.09i)6-s + (2.59 + 2.99i)7-s + (−2.82 − 0.128i)8-s + (−5.37 − 1.57i)9-s + (1.36 − 0.377i)10-s + (1.45 − 2.27i)11-s + (−1.01 + 5.77i)12-s + (1.80 + 1.56i)13-s + (−3.60 − 4.29i)14-s + (−0.417 − 2.90i)15-s + (3.99 + 0.242i)16-s + (−6.50 + 2.97i)17-s + ⋯ |
| L(s) = 1 | + (−0.999 − 0.0151i)2-s + (−0.241 + 1.67i)3-s + (0.999 + 0.0303i)4-s + (−0.429 + 0.125i)5-s + (0.266 − 1.67i)6-s + (0.981 + 1.13i)7-s + (−0.998 − 0.0454i)8-s + (−1.79 − 0.526i)9-s + (0.430 − 0.119i)10-s + (0.440 − 0.684i)11-s + (−0.291 + 1.66i)12-s + (0.500 + 0.433i)13-s + (−0.964 − 1.14i)14-s + (−0.107 − 0.749i)15-s + (0.998 + 0.0606i)16-s + (−1.57 + 0.720i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.125633 - 0.677879i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.125633 - 0.677879i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 + 0.0214i)T \) |
| 5 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (2.91 - 3.80i)T \) |
| good | 3 | \( 1 + (0.417 - 2.90i)T + (-2.87 - 0.845i)T^{2} \) |
| 7 | \( 1 + (-2.59 - 2.99i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-1.45 + 2.27i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.80 - 1.56i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (6.50 - 2.97i)T + (11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-4.78 - 2.18i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (3.38 - 1.54i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (8.17 - 1.17i)T + (29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (2.31 + 0.680i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-7.76 + 2.28i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-4.51 - 0.649i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 + (-6.73 - 7.77i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-0.524 + 0.605i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (0.543 + 3.77i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (8.78 + 13.6i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-5.47 - 8.51i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.80 + 3.95i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (5.70 - 6.58i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-3.96 + 13.5i)T + (-69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (17.0 + 2.45i)T + (85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (2.42 + 8.26i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−10.66621662851208392013751409500, −9.500249309297463271887961747381, −8.938996217392640653888452844758, −8.538341482745470137786436211411, −7.43139203331153063904510692028, −5.98889774372135212547208189194, −5.50211083070277266693564234642, −4.18936390428185761580804198207, −3.35769848794675613947588614985, −1.89736573386009468763285616434,
0.48087830802962938174897551989, 1.44123771062695352579822469736, 2.45403570093161692106988955297, 4.16608095617776281535068356762, 5.56160211387062065060859809197, 6.75276627968232298048742261144, 7.22969352666708549700963128128, 7.72722447478751001740864801504, 8.517495578962263110986039876157, 9.409450177728850767552934071033
