L(s) = 1 | + (1 + i)2-s + 2i·4-s + (0.133 − 0.5i)5-s + (2.13 + 1.23i)7-s + (−2 + 2i)8-s + (0.633 − 0.366i)10-s + (0.5 − 0.133i)11-s + (4.59 + 1.23i)13-s + (0.901 + 3.36i)14-s − 4·16-s − 4·17-s + (−3 + 3i)19-s + (1 + 0.267i)20-s + (0.633 + 0.366i)22-s + (0.401 − 0.232i)23-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + i·4-s + (0.0599 − 0.223i)5-s + (0.806 + 0.465i)7-s + (−0.707 + 0.707i)8-s + (0.200 − 0.115i)10-s + (0.150 − 0.0403i)11-s + (1.27 + 0.341i)13-s + (0.241 + 0.899i)14-s − 16-s − 0.970·17-s + (−0.688 + 0.688i)19-s + (0.223 + 0.0599i)20-s + (0.135 + 0.0780i)22-s + (0.0838 − 0.0483i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0436 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0436 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.53426 + 1.46873i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.53426 + 1.46873i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.133 + 0.5i)T + (-4.33 - 2.5i)T^{2} \) |
| 7 | \( 1 + (-2.13 - 1.23i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.133i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-4.59 - 1.23i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + (3 - 3i)T - 19iT^{2} \) |
| 23 | \( 1 + (-0.401 + 0.232i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.866 + 3.23i)T + (-25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (-0.598 - 1.03i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (7.73 + 7.73i)T + 37iT^{2} \) |
| 41 | \( 1 + (-9.69 + 5.59i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-8.69 + 2.33i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (4.59 - 7.96i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.26 + 2.26i)T + 53iT^{2} \) |
| 59 | \( 1 + (-1.5 + 5.59i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (3.86 + 14.4i)T + (-52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (-1.23 - 0.330i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 10.9iT - 71T^{2} \) |
| 73 | \( 1 + 0.535iT - 73T^{2} \) |
| 79 | \( 1 + (-0.866 + 1.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.16 + 11.7i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + 11.8iT - 89T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.37385124117238564184406445428, −10.86927752403998098214290499492, −9.063486702767007750605709710312, −8.629821109729049739532567959910, −7.65074202273372157566079741528, −6.50152729609205577339568083303, −5.71325454549113683281076716226, −4.64447216272210668371467189189, −3.71251742116378939647714884110, −2.06062063676510350442717570419,
1.28215177371447710200911774828, 2.73817215348547152477752612920, 4.05309217794302992536761349981, 4.84857110397660473702567812082, 6.10506541828453617118233986961, 6.94361858775481856162498027275, 8.374220215361075139299570888491, 9.199098661534823205989293438068, 10.60532063596027598447138417262, 10.85038914651706271754813257857