| L(s) = 1 | + (1.55 + 1.25i)2-s + (0.842 + 3.91i)4-s + (−0.661 + 1.14i)5-s + (4.89 + 8.47i)7-s + (−3.60 + 7.14i)8-s + (−2.47 + 0.952i)10-s + (−6.81 − 11.8i)11-s + (1.13 + 0.654i)13-s + (−3.03 + 19.3i)14-s + (−14.5 + 6.58i)16-s + 0.636i·17-s + 22.9i·19-s + (−5.04 − 1.62i)20-s + (4.22 − 26.9i)22-s + (22.3 + 12.9i)23-s + ⋯ |
| L(s) = 1 | + (0.778 + 0.628i)2-s + (0.210 + 0.977i)4-s + (−0.132 + 0.229i)5-s + (0.699 + 1.21i)7-s + (−0.450 + 0.892i)8-s + (−0.247 + 0.0952i)10-s + (−0.619 − 1.07i)11-s + (0.0872 + 0.0503i)13-s + (−0.216 + 1.38i)14-s + (−0.911 + 0.411i)16-s + 0.0374i·17-s + 1.20i·19-s + (−0.252 − 0.0811i)20-s + (0.192 − 1.22i)22-s + (0.972 + 0.561i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.343 - 0.938i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.343 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.30481 + 1.86756i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30481 + 1.86756i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.55 - 1.25i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (0.661 - 1.14i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-4.89 - 8.47i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (6.81 + 11.8i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-1.13 - 0.654i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 0.636iT - 289T^{2} \) |
| 19 | \( 1 - 22.9iT - 361T^{2} \) |
| 23 | \( 1 + (-22.3 - 12.9i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (6.64 + 11.5i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-18.7 + 32.5i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 51.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (31.8 + 18.3i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-56.7 + 32.7i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-75.9 + 43.8i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 9.23T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-14.5 + 25.2i)T + (-1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-7.53 + 4.35i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (24.3 + 14.0i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 83.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 88.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-22.0 - 38.1i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-64.9 - 112. i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 23.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-1.36 - 2.36i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−12.44302648659325866077162383735, −11.59541458288853607384592892841, −10.82766272671269813533545391680, −9.053079555558960505462592111750, −8.268538278008904391953237744239, −7.33464579492072655184735811108, −5.81700838868926323447240338538, −5.38715151336511312861676881661, −3.76917234967110419121276755448, −2.45537152634157809099252971186,
1.08542952056873030870367155079, 2.79823271790099406768158070857, 4.49655201198496664049860171256, 4.86272423118910144492869059108, 6.64600773696778717746750389428, 7.55095692025542680755307331762, 9.008618502699808896627383248173, 10.33784301052088106326942686017, 10.76367631502263681751587616855, 11.86586377248304966624836499048
