L(s) = 1 | + (−0.5 − 0.866i)2-s + (1.5 − 2.59i)4-s + (−2.5 + 4.33i)5-s − 7·8-s + 5·10-s + (−2.5 − 4.33i)16-s − 14·17-s − 22·19-s + (7.50 + 12.9i)20-s + (−17 + 29.4i)23-s + (−12.5 − 21.6i)25-s + (−1 + 1.73i)31-s + (−16.5 + 28.5i)32-s + (7 + 12.1i)34-s + (11 + 19.0i)38-s + ⋯ |
L(s) = 1 | + (−0.250 − 0.433i)2-s + (0.375 − 0.649i)4-s + (−0.5 + 0.866i)5-s − 0.875·8-s + 0.5·10-s + (−0.156 − 0.270i)16-s − 0.823·17-s − 1.15·19-s + (0.375 + 0.649i)20-s + (−0.739 + 1.28i)23-s + (−0.500 − 0.866i)25-s + (−0.0322 + 0.0558i)31-s + (−0.515 + 0.893i)32-s + (0.205 + 0.356i)34-s + (0.289 + 0.501i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0766113 + 0.164293i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0766113 + 0.164293i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.5 - 4.33i)T \) |
good | 2 | \( 1 + (0.5 + 0.866i)T + (-2 + 3.46i)T^{2} \) |
| 7 | \( 1 + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 14T + 289T^{2} \) |
| 19 | \( 1 + 22T + 361T^{2} \) |
| 23 | \( 1 + (17 - 29.4i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (1 - 1.73i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-7 - 12.1i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 86T + 2.80e3T^{2} \) |
| 59 | \( 1 + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-59 - 102. i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 + (49 + 84.8i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (77 + 133. i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−11.24695729373639647678619778021, −10.57600315855219390884594685293, −9.792468513161543745655948759513, −8.764627998005545138459719887879, −7.59633877458315899320980396190, −6.61915776285840484272313761167, −5.85183929586760904875550215167, −4.32778382847608198057906378047, −3.02723988755840065197137322263, −1.86598458648237018427246833584,
0.07796839221838006038521282421, 2.21964785285620202033777801894, 3.76455212898924814557707386119, 4.72510502641948934612071710553, 6.16462809648141667525249024065, 6.99323642123612158462826833226, 8.223913489044351500887777074324, 8.476978157658496165256371944774, 9.551628077936617551967454490951, 10.87219723725438341815567761614