| L(s) = 1 | + (−1.35 + 0.404i)2-s + (0.578 + 1.63i)3-s + (1.67 − 1.09i)4-s + (−1.08 − 1.23i)5-s + (−1.44 − 1.97i)6-s + (−2.69 − 0.354i)7-s + (−1.82 + 2.16i)8-s + (−2.33 + 1.88i)9-s + (1.96 + 1.23i)10-s + (1.67 + 3.39i)11-s + (2.75 + 2.09i)12-s + (−2.14 − 6.31i)13-s + (3.78 − 0.608i)14-s + (1.38 − 2.47i)15-s + (1.59 − 3.66i)16-s + (1.12 + 1.12i)17-s + ⋯ |
| L(s) = 1 | + (−0.958 + 0.286i)2-s + (0.333 + 0.942i)3-s + (0.836 − 0.548i)4-s + (−0.483 − 0.551i)5-s + (−0.589 − 0.807i)6-s + (−1.01 − 0.133i)7-s + (−0.644 + 0.764i)8-s + (−0.777 + 0.629i)9-s + (0.621 + 0.390i)10-s + (0.504 + 1.02i)11-s + (0.796 + 0.605i)12-s + (−0.594 − 1.75i)13-s + (1.01 − 0.162i)14-s + (0.358 − 0.640i)15-s + (0.398 − 0.917i)16-s + (0.274 + 0.274i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0545 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0545 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.252823 - 0.239391i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.252823 - 0.239391i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.35 - 0.404i)T \) |
| 3 | \( 1 + (-0.578 - 1.63i)T \) |
| good | 5 | \( 1 + (1.08 + 1.23i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (2.69 + 0.354i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (-1.67 - 3.39i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (2.14 + 6.31i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (-1.12 - 1.12i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.200 + 1.00i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (0.716 + 5.44i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (3.39 - 0.222i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (3.29 - 1.90i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.66 + 8.36i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (0.893 + 6.78i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (-4.18 + 2.06i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (-2.15 + 0.578i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (6.62 - 9.91i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (6.59 + 7.51i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (-0.342 - 5.22i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (-5.75 - 2.84i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (2.44 + 5.91i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-4.54 + 10.9i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-3.14 - 11.7i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (1.15 + 1.01i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (12.7 - 5.29i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (6.98 + 4.03i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.33527875823092050498970442318, −9.563793611205203599555957405220, −8.930907566304298691523733811651, −7.984773465143365125997133594389, −7.22939847098780993473548480610, −5.95898962747199564290444157359, −4.96531727359683356573641115573, −3.70259886556753901556456750861, −2.50247880574292823546036194197, −0.24903875837388810500811235149,
1.59477306718281434590612971022, 2.97712390905951355533142603941, 3.66926823716857357440574191540, 6.01093043412197951715410086970, 6.76243124911921698897015510943, 7.34664333757604776406014624436, 8.299243050354130274485181258900, 9.329130360862469725381486574333, 9.646766413959974381082056511089, 11.34044250248176617697995043385
