L(s) = 1 | + (−0.804 − 1.16i)2-s + (−0.706 + 1.87i)4-s + (2.80 + 1.61i)5-s + (1.47 − 2.19i)7-s + (2.74 − 0.682i)8-s + (−0.371 − 4.56i)10-s + (−2.08 + 1.20i)11-s + 3.09i·13-s + (−3.74 + 0.0456i)14-s + (−3.00 − 2.64i)16-s + (1.97 + 3.42i)17-s + (2.33 + 1.35i)19-s + (−5.01 + 4.10i)20-s + (3.08 + 1.46i)22-s + (1.37 − 2.37i)23-s + ⋯ |
L(s) = 1 | + (−0.568 − 0.822i)2-s + (−0.353 + 0.935i)4-s + (1.25 + 0.724i)5-s + (0.558 − 0.829i)7-s + (0.970 − 0.241i)8-s + (−0.117 − 1.44i)10-s + (−0.629 + 0.363i)11-s + 0.858i·13-s + (−0.999 + 0.0122i)14-s + (−0.750 − 0.661i)16-s + (0.479 + 0.830i)17-s + (0.536 + 0.309i)19-s + (−1.12 + 0.917i)20-s + (0.657 + 0.311i)22-s + (0.286 − 0.495i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.33307 - 0.191617i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.33307 - 0.191617i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.804 + 1.16i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.47 + 2.19i)T \) |
good | 5 | \( 1 + (-2.80 - 1.61i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.08 - 1.20i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3.09iT - 13T^{2} \) |
| 17 | \( 1 + (-1.97 - 3.42i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.33 - 1.35i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.37 + 2.37i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2.01iT - 29T^{2} \) |
| 31 | \( 1 + (-1.10 - 1.91i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.30 - 2.48i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2.11T + 41T^{2} \) |
| 43 | \( 1 + 11.5iT - 43T^{2} \) |
| 47 | \( 1 + (3.31 - 5.74i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.23 - 1.29i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-10.6 + 6.13i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7.54 + 4.35i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.01 + 2.89i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6.64T + 71T^{2} \) |
| 73 | \( 1 + (-4.77 - 8.27i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.838 - 1.45i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6.47iT - 83T^{2} \) |
| 89 | \( 1 + (-6.98 + 12.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 1.37T + 97T^{2} \) |
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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
−10.64358571515331230691092654475, −10.14055117292466942596628004305, −9.455284600553046010571637120787, −8.280425276273031131472613647029, −7.38407317198412694676848717434, −6.44518483570827249886735691521, −5.06320089851264087121832587194, −3.86084716016702536201818709191, −2.51912302294638971679461760191, −1.50752135460875631962381851789,
1.16919402483769843206285757080, 2.61750835825301870320645072595, 4.96012856957220992483892965721, 5.41151313975107907339442479156, 6.12490411187292545364130022217, 7.53776032344868970146804980250, 8.295468834934789072641249531780, 9.217166028967177038464190932403, 9.701368317598341834929185010578, 10.69035603623673547890144391593