| L(s) = 1 | + (−1.37 − 0.326i)2-s + (0.0779 − 1.73i)3-s + (1.78 + 0.898i)4-s + (−0.296 − 3.00i)5-s + (−0.672 + 2.35i)6-s + (0.110 + 0.0739i)7-s + (−2.16 − 1.81i)8-s + (−2.98 − 0.269i)9-s + (−0.574 + 4.23i)10-s + (−1.00 − 1.88i)11-s + (1.69 − 3.02i)12-s + (0.0663 − 0.673i)13-s + (−0.128 − 0.137i)14-s + (−5.22 + 0.278i)15-s + (2.38 + 3.21i)16-s + (−0.0442 + 0.106i)17-s + ⋯ |
| L(s) = 1 | + (−0.972 − 0.230i)2-s + (0.0450 − 0.998i)3-s + (0.893 + 0.449i)4-s + (−0.132 − 1.34i)5-s + (−0.274 + 0.961i)6-s + (0.0418 + 0.0279i)7-s + (−0.765 − 0.643i)8-s + (−0.995 − 0.0899i)9-s + (−0.181 + 1.33i)10-s + (−0.304 − 0.569i)11-s + (0.489 − 0.872i)12-s + (0.0184 − 0.186i)13-s + (−0.0342 − 0.0368i)14-s + (−1.34 + 0.0717i)15-s + (0.596 + 0.802i)16-s + (−0.0107 + 0.0259i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0420i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0420i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0137784 - 0.655197i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0137784 - 0.655197i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.37 + 0.326i)T \) |
| 3 | \( 1 + (-0.0779 + 1.73i)T \) |
| good | 5 | \( 1 + (0.296 + 3.00i)T + (-4.90 + 0.975i)T^{2} \) |
| 7 | \( 1 + (-0.110 - 0.0739i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (1.00 + 1.88i)T + (-6.11 + 9.14i)T^{2} \) |
| 13 | \( 1 + (-0.0663 + 0.673i)T + (-12.7 - 2.53i)T^{2} \) |
| 17 | \( 1 + (0.0442 - 0.106i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-2.33 + 1.91i)T + (3.70 - 18.6i)T^{2} \) |
| 23 | \( 1 + (1.27 - 6.40i)T + (-21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-0.984 + 1.84i)T + (-16.1 - 24.1i)T^{2} \) |
| 31 | \( 1 + (4.49 + 4.49i)T + 31iT^{2} \) |
| 37 | \( 1 + (1.32 + 1.09i)T + (7.21 + 36.2i)T^{2} \) |
| 41 | \( 1 + (-1.34 + 6.74i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (2.62 - 8.65i)T + (-35.7 - 23.8i)T^{2} \) |
| 47 | \( 1 + (4.07 - 9.83i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (0.483 + 0.904i)T + (-29.4 + 44.0i)T^{2} \) |
| 59 | \( 1 + (0.185 + 1.87i)T + (-57.8 + 11.5i)T^{2} \) |
| 61 | \( 1 + (4.28 + 14.1i)T + (-50.7 + 33.8i)T^{2} \) |
| 67 | \( 1 + (-11.9 + 3.63i)T + (55.7 - 37.2i)T^{2} \) |
| 71 | \( 1 + (-6.31 - 4.21i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-0.325 - 0.487i)T + (-27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (6.16 - 2.55i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-11.7 + 9.64i)T + (16.1 - 81.4i)T^{2} \) |
| 89 | \( 1 + (-11.3 + 2.25i)T + (82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (-12.8 + 12.8i)T - 97iT^{2} \) |
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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
−11.15888381676093474608366109538, −9.657498440995030055447755047017, −8.960929347170087377602661064763, −8.080165580424680043782196605308, −7.56876193365170033992375378530, −6.25267014639449228916510709769, −5.23817878400641341826610267208, −3.36650452097070350595674968032, −1.82848842910681254516805018113, −0.57826992083058496447746239223,
2.40486948686822854922437510886, 3.49420609913902684601307877204, 5.08243086351153662681318637261, 6.32301540493924493298303780683, 7.16057179382160334476833765000, 8.183181982307961286015927337672, 9.158581430577854696172746412184, 10.21551280891482242827135001795, 10.47500288672792524605353649274, 11.34629748774403399671238020426
