| L(s) = 1 | + (1.06 + 0.514i)2-s + (−0.371 − 0.465i)4-s + (−2.37 + 2.20i)5-s + (−1.38 + 2.39i)7-s + (−0.684 − 2.99i)8-s + (−3.67 + 1.13i)10-s + (−3.47 + 4.36i)11-s + (1.57 + 0.484i)13-s + (−2.71 + 1.84i)14-s + (0.546 − 2.39i)16-s + (−0.555 − 0.515i)17-s + (−0.795 + 2.02i)19-s + (1.90 + 0.287i)20-s + (−5.95 + 2.86i)22-s + (−0.00139 − 0.000210i)23-s + ⋯ |
| L(s) = 1 | + (0.755 + 0.363i)2-s + (−0.185 − 0.232i)4-s + (−1.06 + 0.986i)5-s + (−0.523 + 0.906i)7-s + (−0.241 − 1.06i)8-s + (−1.16 + 0.358i)10-s + (−1.04 + 1.31i)11-s + (0.435 + 0.134i)13-s + (−0.724 + 0.493i)14-s + (0.136 − 0.598i)16-s + (−0.134 − 0.125i)17-s + (−0.182 + 0.465i)19-s + (0.426 + 0.0643i)20-s + (−1.27 + 0.611i)22-s + (−0.000290 − 4.38e−5i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.257362 + 0.858602i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.257362 + 0.858602i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + (-6.54 - 0.330i)T \) |
| good | 2 | \( 1 + (-1.06 - 0.514i)T + (1.24 + 1.56i)T^{2} \) |
| 5 | \( 1 + (2.37 - 2.20i)T + (0.373 - 4.98i)T^{2} \) |
| 7 | \( 1 + (1.38 - 2.39i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.47 - 4.36i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.57 - 0.484i)T + (10.7 + 7.32i)T^{2} \) |
| 17 | \( 1 + (0.555 + 0.515i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (0.795 - 2.02i)T + (-13.9 - 12.9i)T^{2} \) |
| 23 | \( 1 + (0.00139 + 0.000210i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-3.87 + 2.63i)T + (10.5 - 26.9i)T^{2} \) |
| 31 | \( 1 + (-0.301 - 4.02i)T + (-30.6 + 4.62i)T^{2} \) |
| 37 | \( 1 + (0.999 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.30 + 3.03i)T + (25.5 + 32.0i)T^{2} \) |
| 47 | \( 1 + (-4.90 - 6.14i)T + (-10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (-3.76 + 1.16i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.811 + 3.55i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (0.639 - 8.52i)T + (-60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (-3.43 + 8.73i)T + (-49.1 - 45.5i)T^{2} \) |
| 71 | \( 1 + (6.83 - 1.02i)T + (67.8 - 20.9i)T^{2} \) |
| 73 | \( 1 + (10.6 + 3.27i)T + (60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 + (3.26 - 5.66i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.71 - 3.21i)T + (30.3 + 77.2i)T^{2} \) |
| 89 | \( 1 + (-7.13 - 4.86i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (11.0 - 13.8i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−12.02486186249445900066542009000, −10.71870375146087633917559208911, −10.04463345316520001107092495876, −8.940619629697486386189240083298, −7.68770453371388233594816365873, −6.87157361122769010678279898567, −5.95017287301898335158924914994, −4.83398156145609537220165230610, −3.79192571197377179869072272896, −2.62945005462404411707973229926,
0.46091883495946978966494368418, 3.03530105613337810462661596644, 3.90009640268016518344879270762, 4.74476561007061722086423945158, 5.80914695953111181585933638059, 7.35403338681790742839647067532, 8.343349700435143336857892607691, 8.744739132327721318640915090663, 10.34351527882944631825932747242, 11.20297572950925485798903654386
