L(s) = 1 | + (−0.420 + 1.35i)2-s + (2.28 − 0.946i)3-s + (−1.64 − 1.13i)4-s + (0.446 − 1.07i)5-s + (0.316 + 3.48i)6-s + (−0.707 − 0.707i)7-s + (2.22 − 1.74i)8-s + (2.20 − 2.20i)9-s + (1.26 + 1.05i)10-s + (2.89 + 1.19i)11-s + (−4.83 − 1.03i)12-s + (1.10 + 2.65i)13-s + (1.25 − 0.657i)14-s − 2.88i·15-s + (1.41 + 3.73i)16-s − 3.12i·17-s + ⋯ |
L(s) = 1 | + (−0.297 + 0.954i)2-s + (1.31 − 0.546i)3-s + (−0.823 − 0.567i)4-s + (0.199 − 0.481i)5-s + (0.129 + 1.42i)6-s + (−0.267 − 0.267i)7-s + (0.786 − 0.616i)8-s + (0.734 − 0.734i)9-s + (0.400 + 0.333i)10-s + (0.871 + 0.361i)11-s + (−1.39 − 0.299i)12-s + (0.305 + 0.737i)13-s + (0.334 − 0.175i)14-s − 0.744i·15-s + (0.354 + 0.934i)16-s − 0.758i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.48180 + 0.191665i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.48180 + 0.191665i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.420 - 1.35i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 3 | \( 1 + (-2.28 + 0.946i)T + (2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (-0.446 + 1.07i)T + (-3.53 - 3.53i)T^{2} \) |
| 11 | \( 1 + (-2.89 - 1.19i)T + (7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (-1.10 - 2.65i)T + (-9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + 3.12iT - 17T^{2} \) |
| 19 | \( 1 + (1.63 + 3.95i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (1.37 - 1.37i)T - 23iT^{2} \) |
| 29 | \( 1 + (8.15 - 3.37i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 2.07T + 31T^{2} \) |
| 37 | \( 1 + (2.47 - 5.96i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (5.68 - 5.68i)T - 41iT^{2} \) |
| 43 | \( 1 + (-5.63 - 2.33i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + 8.41iT - 47T^{2} \) |
| 53 | \( 1 + (12.4 + 5.16i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-2.20 + 5.32i)T + (-41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (-4.89 + 2.02i)T + (43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (-8.29 + 3.43i)T + (47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (-6.35 - 6.35i)T + 71iT^{2} \) |
| 73 | \( 1 + (10.6 - 10.6i)T - 73iT^{2} \) |
| 79 | \( 1 + 5.81iT - 79T^{2} \) |
| 83 | \( 1 + (1.91 + 4.62i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-5.79 - 5.79i)T + 89iT^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−12.88709313463125116171581893685, −11.32471952101022801504753017670, −9.640268155984212018461125683629, −9.164878183352105001720478028976, −8.402341960555268608076160067882, −7.24485934754130908292217007268, −6.64562693086589951070332328279, −4.99475732920135165807201801437, −3.64169716848360266283034707439, −1.63372582738785931005600295559,
2.09967920408832612637111031638, 3.35250040497967046038613686707, 4.02766970452384769424172380369, 5.95373136539693982063021914379, 7.71911397627063791466536126251, 8.628657093929688557906532704846, 9.292013894446044270536117356362, 10.22830564268049249689419073921, 10.94557748636939011998116246031, 12.27137396885066904687880781729