| L(s) = 1 | + (0.766 − 0.642i)2-s + (0.503 − 1.65i)3-s + (0.173 − 0.984i)4-s + (−4.01 + 1.46i)5-s + (−0.679 − 1.59i)6-s + (−0.173 − 0.984i)7-s + (−0.500 − 0.866i)8-s + (−2.49 − 1.66i)9-s + (−2.13 + 3.70i)10-s + (−4.81 − 1.75i)11-s + (−1.54 − 0.783i)12-s + (0.991 + 0.832i)13-s + (−0.766 − 0.642i)14-s + (0.398 + 7.39i)15-s + (−0.939 − 0.342i)16-s + (2.73 − 4.73i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (0.290 − 0.956i)3-s + (0.0868 − 0.492i)4-s + (−1.79 + 0.653i)5-s + (−0.277 − 0.650i)6-s + (−0.0656 − 0.372i)7-s + (−0.176 − 0.306i)8-s + (−0.830 − 0.556i)9-s + (−0.675 + 1.17i)10-s + (−1.45 − 0.528i)11-s + (−0.445 − 0.226i)12-s + (0.275 + 0.230i)13-s + (−0.204 − 0.171i)14-s + (0.102 + 1.90i)15-s + (−0.234 − 0.0855i)16-s + (0.662 − 1.14i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0210582 - 0.927520i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0210582 - 0.927520i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 3 | \( 1 + (-0.503 + 1.65i)T \) |
| 7 | \( 1 + (0.173 + 0.984i)T \) |
| good | 5 | \( 1 + (4.01 - 1.46i)T + (3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (4.81 + 1.75i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.991 - 0.832i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.73 + 4.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.05 - 1.82i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.926 + 5.25i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.526 + 0.441i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.971 - 5.51i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-4.27 + 7.40i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.19 - 1.00i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (2.26 + 0.825i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.903 - 5.12i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 5.73T + 53T^{2} \) |
| 59 | \( 1 + (11.4 - 4.18i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.404 + 2.29i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (7.73 + 6.49i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.20 + 3.81i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (7.78 + 13.4i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.893 - 0.749i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-9.57 + 8.03i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (2.39 + 4.15i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.60 - 3.49i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.06723722192093253237135957108, −10.50661300460875313462529232785, −8.836964482101657860641050911807, −7.72522310084897241154392054586, −7.41725356171218050937728264907, −6.21239030393081115682185043368, −4.77196339009058397381113709132, −3.42386296510020151657004304057, −2.78157490859410933300398685448, −0.49507811634385625052456200431,
3.02259204875884687890143779244, 3.94473473845356047946598202133, 4.84202837423251862958301386806, 5.63891782705092798226284919318, 7.49193566717915960669576759952, 8.034486209795618539867291843831, 8.742397166471614271313142495863, 10.00305428184874531932955658009, 11.08799283829938850166974539145, 11.79398315715951908186484821695
