L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.358 − 0.621i)5-s + 0.999i·8-s + (0.621 + 0.358i)10-s + (2.59 + 1.5i)11-s − 2.44i·13-s + (−0.5 − 0.866i)16-s + (−2.95 + 5.12i)17-s + (5.12 − 2.95i)19-s − 0.717·20-s − 3·22-s + (−3.67 + 2.12i)23-s + (2.24 − 3.88i)25-s + (1.22 + 2.12i)26-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.160 − 0.277i)5-s + 0.353i·8-s + (0.196 + 0.113i)10-s + (0.783 + 0.452i)11-s − 0.679i·13-s + (−0.125 − 0.216i)16-s + (−0.717 + 1.24i)17-s + (1.17 − 0.678i)19-s − 0.160·20-s − 0.639·22-s + (−0.766 + 0.442i)23-s + (0.448 − 0.776i)25-s + (0.240 + 0.416i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0285i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.16542 + 0.0166118i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.16542 + 0.0166118i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.358 + 0.621i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.59 - 1.5i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 + (2.95 - 5.12i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.12 + 2.95i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.67 - 2.12i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 7.24iT - 29T^{2} \) |
| 31 | \( 1 + (-7.86 - 4.54i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.121 - 0.210i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 + 0.242T + 43T^{2} \) |
| 47 | \( 1 + (2.95 + 5.12i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.27 - 3.62i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.03 + 6.98i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.878 - 0.507i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 1.75iT - 71T^{2} \) |
| 73 | \( 1 + (-1.24 - 0.717i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.37 - 2.38i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6.63T + 83T^{2} \) |
| 89 | \( 1 + (-5.19 - 9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 13.5iT - 97T^{2} \) |
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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
−9.982045182737992740675596881247, −9.297389975734942590272903998907, −8.376245877538508571028464184109, −7.78715647126560771656769761031, −6.71705452042696099357111012116, −6.02251739358501573278288847880, −4.87210561872456002430514270402, −3.87337227124576847480095321658, −2.37428352175997542567851884347, −0.921958499226523987839982984909,
1.05983981774800557796664473423, 2.52317465397626077753882305924, 3.57609637638814789604612707075, 4.62993750930474617695084002409, 5.96276170963510567854847921548, 6.90772853517415410480401852979, 7.58116353143007913551541015783, 8.662939085809014831329246771016, 9.313571705693782483087613265571, 10.00549538128055183969981614503