| L(s) = 1 | + (0.347 + 1.37i)2-s + (0.654 + 0.755i)3-s + (−1.75 + 0.952i)4-s + (0.441 + 3.07i)5-s + (−0.808 + 1.16i)6-s + (−1.53 + 3.35i)7-s + (−1.91 − 2.08i)8-s + (−0.142 + 0.989i)9-s + (−4.05 + 1.67i)10-s + (0.0724 − 0.246i)11-s + (−1.87 − 0.705i)12-s + (5.35 − 2.44i)13-s + (−5.13 − 0.935i)14-s + (−2.03 + 2.34i)15-s + (2.18 − 3.34i)16-s + (1.69 − 2.64i)17-s + ⋯ |
| L(s) = 1 | + (0.245 + 0.969i)2-s + (0.378 + 0.436i)3-s + (−0.879 + 0.476i)4-s + (0.197 + 1.37i)5-s + (−0.330 + 0.473i)6-s + (−0.579 + 1.26i)7-s + (−0.677 − 0.735i)8-s + (−0.0474 + 0.329i)9-s + (−1.28 + 0.528i)10-s + (0.0218 − 0.0743i)11-s + (−0.540 − 0.203i)12-s + (1.48 − 0.678i)13-s + (−1.37 − 0.250i)14-s + (−0.524 + 0.605i)15-s + (0.546 − 0.837i)16-s + (0.411 − 0.640i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.122594 - 1.57106i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.122594 - 1.57106i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.347 - 1.37i)T \) |
| 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (1.99 + 4.36i)T \) |
| good | 5 | \( 1 + (-0.441 - 3.07i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (1.53 - 3.35i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.0724 + 0.246i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-5.35 + 2.44i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-1.69 + 2.64i)T + (-7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (1.92 + 2.99i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (4.01 - 6.24i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-7.88 - 6.83i)T + (4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.699 + 4.86i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.377 - 2.62i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (4.72 - 4.09i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 12.3iT - 47T^{2} \) |
| 53 | \( 1 + (2.05 - 4.49i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-2.01 - 4.40i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-1.41 + 1.62i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (2.74 + 9.34i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (2.77 + 9.46i)T + (-59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-9.51 + 6.11i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (3.04 + 6.67i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (2.47 + 0.356i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-12.0 + 10.4i)T + (12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-14.6 + 2.10i)T + (93.0 - 27.3i)T^{2} \) |
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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
−11.05153366839743559025717393278, −10.27658552659291038964152365676, −9.216304901106118936759262410827, −8.643118339361252887363850425810, −7.62767444697946112494475766729, −6.40621214955173241913175205769, −6.09678074554071023307050501874, −4.86433716387739401415610583626, −3.33245514229457688335798650373, −2.85636278159812233510728172913,
0.866226878368712261972338676344, 1.82902008066488040500871899600, 3.79436449036467346099327892590, 4.06218054341909587703024017219, 5.56483882495785885136265388889, 6.49861579447299195300931512713, 8.057188352578457106953303077570, 8.584027880256013430827238642527, 9.671298236702419661907260225978, 10.14421764195939761628158232663
