| L(s) = 1 | + (−1.21 + 1.48i)2-s + (−0.996 − 0.0804i)3-s + (−0.337 − 1.65i)4-s + (−0.0596 − 0.491i)5-s + (1.33 − 1.38i)6-s + (0.351 − 0.222i)7-s + (−0.533 − 0.279i)8-s + (0.987 + 0.160i)9-s + (0.802 + 0.507i)10-s + (−0.655 + 0.106i)11-s + (0.203 + 1.67i)12-s + (3.60 − 0.196i)13-s + (−0.0962 + 0.792i)14-s + (0.0199 + 0.494i)15-s + (4.16 − 1.77i)16-s + (1.53 − 0.972i)17-s + ⋯ |
| L(s) = 1 | + (−0.858 + 1.05i)2-s + (−0.575 − 0.0464i)3-s + (−0.168 − 0.826i)4-s + (−0.0266 − 0.219i)5-s + (0.542 − 0.565i)6-s + (0.132 − 0.0840i)7-s + (−0.188 − 0.0989i)8-s + (0.329 + 0.0534i)9-s + (0.253 + 0.160i)10-s + (−0.197 + 0.0321i)11-s + (0.0586 + 0.483i)12-s + (0.998 − 0.0544i)13-s + (−0.0257 + 0.211i)14-s + (0.00514 + 0.127i)15-s + (1.04 − 0.443i)16-s + (0.372 − 0.235i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.363 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.389145 + 0.569680i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.389145 + 0.569680i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.996 + 0.0804i)T \) |
| 13 | \( 1 + (-3.60 + 0.196i)T \) |
| good | 2 | \( 1 + (1.21 - 1.48i)T + (-0.400 - 1.95i)T^{2} \) |
| 5 | \( 1 + (0.0596 + 0.491i)T + (-4.85 + 1.19i)T^{2} \) |
| 7 | \( 1 + (-0.351 + 0.222i)T + (3.00 - 6.32i)T^{2} \) |
| 11 | \( 1 + (0.655 - 0.106i)T + (10.4 - 3.48i)T^{2} \) |
| 17 | \( 1 + (-1.53 + 0.972i)T + (7.28 - 15.3i)T^{2} \) |
| 19 | \( 1 + (3.32 - 5.75i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.79 - 3.10i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.20 - 3.92i)T + (-5.80 - 28.4i)T^{2} \) |
| 31 | \( 1 + (-9.76 - 2.40i)T + (27.4 + 14.4i)T^{2} \) |
| 37 | \( 1 + (1.40 + 4.85i)T + (-31.2 + 19.7i)T^{2} \) |
| 41 | \( 1 + (11.9 + 0.960i)T + (40.4 + 6.57i)T^{2} \) |
| 43 | \( 1 + (2.78 - 9.59i)T + (-36.3 - 22.9i)T^{2} \) |
| 47 | \( 1 + (-7.97 - 7.06i)T + (5.66 + 46.6i)T^{2} \) |
| 53 | \( 1 + (-7.81 - 4.10i)T + (30.1 + 43.6i)T^{2} \) |
| 59 | \( 1 + (5.70 + 2.43i)T + (40.8 + 42.5i)T^{2} \) |
| 61 | \( 1 + (-0.350 + 8.69i)T + (-60.8 - 4.90i)T^{2} \) |
| 67 | \( 1 + (-0.0907 + 0.444i)T + (-61.6 - 26.2i)T^{2} \) |
| 71 | \( 1 + (0.299 + 0.630i)T + (-44.9 + 54.9i)T^{2} \) |
| 73 | \( 1 + (-1.99 + 5.26i)T + (-54.6 - 48.4i)T^{2} \) |
| 79 | \( 1 + (-4.53 - 4.01i)T + (9.52 + 78.4i)T^{2} \) |
| 83 | \( 1 + (-4.76 - 6.90i)T + (-29.4 + 77.6i)T^{2} \) |
| 89 | \( 1 + (-2.35 - 4.07i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.4 - 7.82i)T + (26.9 + 93.1i)T^{2} \) |
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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
−10.91489260164939176403291890268, −10.20251120617546631239206197477, −9.188807722955373232686688366995, −8.352089665888971026105062745131, −7.67042575243080633139051712611, −6.60658177288337987689616097137, −5.96116598298391750845290321913, −4.91042185066949229693855657771, −3.43533890919691199824777253839, −1.19236398356986542476964896868,
0.71307619989222435072218932842, 2.20793964655731344053905833846, 3.45195184683116849182347032131, 4.84636120021376290742194076884, 6.05983167491525432261636618940, 6.98415269976241386696133954890, 8.464994700178614720875685147867, 8.828752719905703326905916172099, 10.26305399202747115432779533678, 10.40706850023929550180312192778
