| L(s) = 1 | + (1.22 + 0.707i)2-s + (0.999 + 1.73i)4-s + (1.73 + i)5-s + 2.82i·8-s + (1.41 + 2.44i)10-s + (2.44 − 1.41i)11-s + 12.7·13-s + (−2.00 + 3.46i)16-s + (19.0 − 11i)17-s + (2.82 − 4.89i)19-s + 3.99i·20-s + 4·22-s + (2.44 + 1.41i)23-s + (−10.5 − 18.1i)25-s + (15.5 + 9i)26-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.346 + 0.200i)5-s + 0.353i·8-s + (0.141 + 0.244i)10-s + (0.222 − 0.128i)11-s + 0.979·13-s + (−0.125 + 0.216i)16-s + (1.12 − 0.647i)17-s + (0.148 − 0.257i)19-s + 0.199i·20-s + 0.181·22-s + (0.106 + 0.0614i)23-s + (−0.419 − 0.727i)25-s + (0.599 + 0.346i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 - 0.772i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.171865698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.171865698\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.22 - 0.707i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (-1.73 - i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (-2.44 + 1.41i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 12.7T + 169T^{2} \) |
| 17 | \( 1 + (-19.0 + 11i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-2.82 + 4.89i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-2.44 - 1.41i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 35.3iT - 841T^{2} \) |
| 31 | \( 1 + (-16.9 - 29.3i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (32 - 55.4i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 20iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 44T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-58.8 - 34i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (15.9 - 9.19i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-86.6 + 50i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (26.1 - 45.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (60 + 103. i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 8.48iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-37.4 - 64.9i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (46 - 79.6i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 112iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (17.3 + 10i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 26.8T + 9.40e3T^{2} \) |
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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
−10.16223363300567210135608968542, −9.103835149155926012969823062114, −8.323746838970727621342618836254, −7.33034928705885765938929132050, −6.51880972295573740426591474502, −5.70832699986077464683601958411, −4.85358789584483933468507156903, −3.66417359431879469082221265676, −2.81225263068326825393729991674, −1.25302691378317648851690875911,
1.03619272163715768737898668366, 2.18853702664723317077055047215, 3.54306590059391584366428481425, 4.23996762780756097566360559614, 5.65912498185803542703558158295, 5.91688379292202082806817051610, 7.19394882781846853355160383691, 8.131767456479474930291048980327, 9.140070939591639760676780731424, 9.937787355598866772007547268037
