| L(s) = 1 | + (−1.89 − 0.949i)2-s + (−0.540 + 1.64i)3-s + (1.47 + 1.98i)4-s + (−1.11 − 3.72i)5-s + (2.58 − 2.59i)6-s + (−1.32 − 3.06i)7-s + (−0.175 − 0.992i)8-s + (−2.41 − 1.77i)9-s + (−1.43 + 8.11i)10-s + (0.736 − 0.174i)11-s + (−4.06 + 1.35i)12-s + (−2.07 − 1.36i)13-s + (−0.410 + 7.04i)14-s + (6.74 + 0.180i)15-s + (0.808 − 2.70i)16-s + (−0.700 + 0.255i)17-s + ⋯ |
| L(s) = 1 | + (−1.33 − 0.671i)2-s + (−0.312 + 0.949i)3-s + (0.739 + 0.993i)4-s + (−0.499 − 1.66i)5-s + (1.05 − 1.06i)6-s + (−0.499 − 1.15i)7-s + (−0.0618 − 0.350i)8-s + (−0.804 − 0.593i)9-s + (−0.452 + 2.56i)10-s + (0.222 − 0.0526i)11-s + (−1.17 + 0.392i)12-s + (−0.576 − 0.378i)13-s + (−0.109 + 1.88i)14-s + (1.74 + 0.0464i)15-s + (0.202 − 0.675i)16-s + (−0.170 + 0.0618i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 + 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.140214 - 0.289678i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.140214 - 0.289678i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.540 - 1.64i)T \) |
| good | 2 | \( 1 + (1.89 + 0.949i)T + (1.19 + 1.60i)T^{2} \) |
| 5 | \( 1 + (1.11 + 3.72i)T + (-4.17 + 2.74i)T^{2} \) |
| 7 | \( 1 + (1.32 + 3.06i)T + (-4.80 + 5.09i)T^{2} \) |
| 11 | \( 1 + (-0.736 + 0.174i)T + (9.82 - 4.93i)T^{2} \) |
| 13 | \( 1 + (2.07 + 1.36i)T + (5.14 + 11.9i)T^{2} \) |
| 17 | \( 1 + (0.700 - 0.255i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (-4.21 - 1.53i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (0.905 - 2.09i)T + (-15.7 - 16.7i)T^{2} \) |
| 29 | \( 1 + (0.0269 + 0.463i)T + (-28.8 + 3.36i)T^{2} \) |
| 31 | \( 1 + (-3.91 + 0.457i)T + (30.1 - 7.14i)T^{2} \) |
| 37 | \( 1 + (3.64 + 3.05i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (-4.37 + 2.19i)T + (24.4 - 32.8i)T^{2} \) |
| 43 | \( 1 + (4.07 + 4.32i)T + (-2.50 + 42.9i)T^{2} \) |
| 47 | \( 1 + (-10.6 - 1.24i)T + (45.7 + 10.8i)T^{2} \) |
| 53 | \( 1 + (-5.75 + 9.96i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.03 + 0.956i)T + (52.7 + 26.4i)T^{2} \) |
| 61 | \( 1 + (-0.159 + 0.214i)T + (-17.4 - 58.4i)T^{2} \) |
| 67 | \( 1 + (-0.111 + 1.91i)T + (-66.5 - 7.77i)T^{2} \) |
| 71 | \( 1 + (-1.17 + 6.65i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-1.37 - 7.81i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-3.85 - 1.93i)T + (47.1 + 63.3i)T^{2} \) |
| 83 | \( 1 + (2.27 + 1.14i)T + (49.5 + 66.5i)T^{2} \) |
| 89 | \( 1 + (0.935 + 5.30i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (2.72 - 9.08i)T + (-81.0 - 53.3i)T^{2} \) |
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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
−13.86960775547112311492034889629, −12.40437001418114739915711363037, −11.58407407845842851487613036513, −10.35667382222186008830559809558, −9.607704378078884904767108955849, −8.737383098361242387688204279802, −7.59332489173502304751349434668, −5.21343699202455759598456494930, −3.82256727106593797596208455724, −0.65313545517410132349936418840,
2.68332026846822166023518666647, 6.07624095970256970652784133149, 6.89366701483700780734480304809, 7.66929643787320153480979542343, 8.952969005443303991695080900355, 10.20335504741230260896978766580, 11.39095742699167537741413250596, 12.27819713129108696983468018758, 13.94327115924738615636121362752, 15.06547416552663174001359251990
