L(s) = 1 | + (−1.20 − 2.09i)2-s + (−1.91 + 3.31i)4-s + (−0.292 − 0.507i)5-s + 4.41·8-s + (−0.707 + 1.22i)10-s + (−1 + 1.73i)11-s − 5.41·13-s + (−1.49 − 2.59i)16-s + (−3.12 + 5.40i)17-s + (1.41 + 2.44i)19-s + 2.24·20-s + 4.82·22-s + (1.82 + 3.16i)23-s + (2.32 − 4.03i)25-s + (6.53 + 11.3i)26-s + ⋯ |
L(s) = 1 | + (−0.853 − 1.47i)2-s + (−0.957 + 1.65i)4-s + (−0.130 − 0.226i)5-s + 1.56·8-s + (−0.223 + 0.387i)10-s + (−0.301 + 0.522i)11-s − 1.50·13-s + (−0.374 − 0.649i)16-s + (−0.757 + 1.31i)17-s + (0.324 + 0.561i)19-s + 0.501·20-s + 1.02·22-s + (0.381 + 0.660i)23-s + (0.465 − 0.806i)25-s + (1.28 + 2.22i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.335371 + 0.103049i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.335371 + 0.103049i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (1.20 + 2.09i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (0.292 + 0.507i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 5.41T + 13T^{2} \) |
| 17 | \( 1 + (3.12 - 5.40i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.41 - 2.44i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.82 - 3.16i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 1.17T + 29T^{2} \) |
| 31 | \( 1 + (3.41 - 5.91i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2.24T + 41T^{2} \) |
| 43 | \( 1 + 5.65T + 43T^{2} \) |
| 47 | \( 1 + (1.41 + 2.44i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.41 + 5.91i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.87 - 3.25i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.82 - 4.89i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + (2.94 - 5.10i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.17 + 2.02i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 + (-2.87 - 4.98i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 5.41T + 97T^{2} \) |
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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
−11.08442227698367814256430054822, −10.22338766037497619362800852580, −9.732772895365562233234470000352, −8.691570187098303785888912431052, −7.976962911792102249882217834644, −6.83791026211567828557377316255, −5.12590229259933459034359106841, −4.01591178117714525488973306410, −2.75319720552970535818630212536, −1.63312452912655866259965414681,
0.28796859418613877262425101429, 2.74363793057589466798992633355, 4.74158837045971618014781920286, 5.47734263462445795433573042490, 6.78546102473409594983307933964, 7.24270604105199894833649870860, 8.135973061998228593406531017041, 9.180747058936178411320339737217, 9.661443799096982677575971547619, 10.80727750820892053434204810990