L(s) = 1 | + (−0.173 − 0.984i)3-s + (−3.37 + 2.82i)5-s + (−1.97 − 1.14i)7-s + (−0.939 + 0.342i)9-s + (0.618 − 0.357i)11-s + (−1.61 − 0.285i)13-s + (3.37 + 2.82i)15-s + (4.33 + 1.57i)17-s + (4.33 − 0.407i)19-s + (−0.779 + 2.14i)21-s + (4.29 − 5.12i)23-s + (2.49 − 14.1i)25-s + (0.5 + 0.866i)27-s + (0.0802 + 0.220i)29-s + (0.525 − 0.910i)31-s + ⋯ |
L(s) = 1 | + (−0.100 − 0.568i)3-s + (−1.50 + 1.26i)5-s + (−0.746 − 0.430i)7-s + (−0.313 + 0.114i)9-s + (0.186 − 0.107i)11-s + (−0.448 − 0.0790i)13-s + (0.870 + 0.730i)15-s + (1.05 + 0.382i)17-s + (0.995 − 0.0934i)19-s + (−0.170 + 0.467i)21-s + (0.895 − 1.06i)23-s + (0.499 − 2.83i)25-s + (0.0962 + 0.166i)27-s + (0.0149 + 0.0409i)29-s + (0.0944 − 0.163i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.846902 - 0.321153i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.846902 - 0.321153i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 + (-4.33 + 0.407i)T \) |
good | 5 | \( 1 + (3.37 - 2.82i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (1.97 + 1.14i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.618 + 0.357i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.61 + 0.285i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-4.33 - 1.57i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-4.29 + 5.12i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.0802 - 0.220i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.525 + 0.910i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 8.20iT - 37T^{2} \) |
| 41 | \( 1 + (-3.04 + 0.537i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-3.07 - 3.66i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-2.76 - 7.59i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (8.01 - 9.55i)T + (-9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-8.13 - 2.95i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.25 - 1.04i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (1.95 - 0.710i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-9.45 + 7.93i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (0.0741 + 0.420i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (2.98 + 16.9i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (8.29 + 4.78i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-9.14 - 1.61i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (0.956 - 2.62i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−10.24991246909775935782802989015, −9.135872371463534059437958497081, −7.890601783387428487785677491340, −7.46344368995273923126090437160, −6.79978191717531634447438004727, −5.94251769270975009223979359917, −4.43550407952945028980933912778, −3.40624461979813542861791190504, −2.78973812840867759189834442706, −0.61249475515057199937431070713,
0.941418509310169785371672474120, 3.16429490390782731637759386384, 3.82097829813550669217116849748, 4.98749924677122037971597581494, 5.42457460648152042949538407949, 6.96049029782471249273389873793, 7.77632927036858934381392724660, 8.570188951522687062235593437882, 9.425542678620197944629697795001, 9.862278903100811825651258117813