L(s) = 1 | + (−0.759 + 1.31i)2-s + (−0.447 + 2.96i)3-s + (0.845 + 1.46i)4-s + (−0.681 + 1.18i)5-s + (−3.56 − 2.84i)6-s + (0.736 − 6.96i)7-s − 8.64·8-s + (−8.59 − 2.65i)9-s + (−1.03 − 1.79i)10-s + (−2.96 − 5.12i)11-s + (−4.72 + 1.85i)12-s − 13·13-s + (8.60 + 6.25i)14-s + (−3.19 − 2.54i)15-s + (3.19 − 5.52i)16-s + (−8.60 + 4.96i)17-s + ⋯ |
L(s) = 1 | + (−0.379 + 0.658i)2-s + (−0.149 + 0.988i)3-s + (0.211 + 0.365i)4-s + (−0.136 + 0.236i)5-s + (−0.594 − 0.473i)6-s + (0.105 − 0.994i)7-s − 1.08·8-s + (−0.955 − 0.295i)9-s + (−0.103 − 0.179i)10-s + (−0.269 − 0.466i)11-s + (−0.393 + 0.154i)12-s − 13-s + (0.614 + 0.447i)14-s + (−0.213 − 0.169i)15-s + (0.199 − 0.345i)16-s + (−0.506 + 0.292i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.155831 - 0.188987i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.155831 - 0.188987i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.447 - 2.96i)T \) |
| 7 | \( 1 + (-0.736 + 6.96i)T \) |
| 13 | \( 1 + 13T \) |
good | 2 | \( 1 + (0.759 - 1.31i)T + (-2 - 3.46i)T^{2} \) |
| 5 | \( 1 + (0.681 - 1.18i)T + (-12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (2.96 + 5.12i)T + (-60.5 + 104. i)T^{2} \) |
| 17 | \( 1 + (8.60 - 4.96i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (7.13 + 4.11i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (2.23 + 1.29i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 21.1iT - 841T^{2} \) |
| 31 | \( 1 + (9.34 - 5.39i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (28.5 + 16.4i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 16.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + 3.30T + 1.84e3T^{2} \) |
| 47 | \( 1 + (32.9 - 57.0i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-58.6 + 33.8i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-8.56 - 14.8i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (9.60 - 16.6i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (88.3 - 50.9i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 27.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + (87.1 - 50.2i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-28.4 + 49.2i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 104.T + 6.88e3T^{2} \) |
| 89 | \( 1 + (43.7 - 75.7i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 27.8iT - 9.40e3T^{2} \) |
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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
−12.19044450928828629720077826331, −11.12784140660288665906873771409, −10.53303828475323219380283964078, −9.397660428190489561949443393768, −8.511951465781669077944992627066, −7.46974879663664358217016588797, −6.62971427178075729860103254489, −5.31331486483692013398724063294, −4.05218419246290389171906160867, −2.93381652437885093446290023516,
0.13014537847348308825329840861, 1.90752719284588504260447447401, 2.69055752671978665171910562493, 4.97777306968923794054643457076, 6.00266274276197863582660946951, 7.00773263263764943757197304195, 8.230373501948830702908470326324, 9.102699727720250357562534929327, 10.11857680364442485608103662616, 11.17701868609049974767781157846