| L(s) = 1 | + (0.956 + 1.06i)2-s + (0.819 − 1.52i)3-s + (−0.00470 + 0.0447i)4-s + (−2.07 + 0.840i)5-s + (2.40 − 0.589i)6-s + (1.92 − 3.32i)7-s + (2.26 − 1.64i)8-s + (−1.65 − 2.50i)9-s + (−2.87 − 1.39i)10-s + (3.49 + 3.88i)11-s + (0.0644 + 0.0438i)12-s + (−2.25 + 2.49i)13-s + (5.37 − 1.14i)14-s + (−0.415 + 3.85i)15-s + (3.99 + 0.849i)16-s + (−3.90 + 2.84i)17-s + ⋯ |
| L(s) = 1 | + (0.676 + 0.751i)2-s + (0.473 − 0.880i)3-s + (−0.00235 + 0.0223i)4-s + (−0.926 + 0.375i)5-s + (0.982 − 0.240i)6-s + (0.725 − 1.25i)7-s + (0.799 − 0.580i)8-s + (−0.552 − 0.833i)9-s + (−0.909 − 0.442i)10-s + (1.05 + 1.17i)11-s + (0.0186 + 0.0126i)12-s + (−0.624 + 0.693i)13-s + (1.43 − 0.305i)14-s + (−0.107 + 0.994i)15-s + (0.999 + 0.212i)16-s + (−0.948 + 0.688i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.240i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.84353 - 0.225060i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.84353 - 0.225060i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.819 + 1.52i)T \) |
| 5 | \( 1 + (2.07 - 0.840i)T \) |
| good | 2 | \( 1 + (-0.956 - 1.06i)T + (-0.209 + 1.98i)T^{2} \) |
| 7 | \( 1 + (-1.92 + 3.32i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.49 - 3.88i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (2.25 - 2.49i)T + (-1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (3.90 - 2.84i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (1.90 - 1.38i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-4.56 + 0.969i)T + (21.0 - 9.35i)T^{2} \) |
| 29 | \( 1 + (6.71 - 2.98i)T + (19.4 - 21.5i)T^{2} \) |
| 31 | \( 1 + (-0.593 - 0.264i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-0.315 - 0.970i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (2.50 - 2.78i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (2.34 - 4.06i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-10.9 + 4.85i)T + (31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (-2.92 - 2.12i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (5.20 - 5.78i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (4.40 + 4.89i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (0.930 + 0.414i)T + (44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (2.82 + 2.04i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (1.38 - 4.25i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-1.49 + 0.667i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (1.18 + 11.2i)T + (-81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 + (0.935 - 2.87i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-14.8 + 6.59i)T + (64.9 - 72.0i)T^{2} \) |
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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
−12.44139323685249572918084101273, −11.42075777171663168670700728976, −10.44342923194477316601453755580, −9.000371105609393115991331465524, −7.63947091637551385752786971126, −7.11915390519570524485044524321, −6.52799381527383407881793726937, −4.57172063322240159082039538819, −3.94101471928783613988302843246, −1.61629563176546933172111785521,
2.47945252674134996962656658870, 3.57052511276764969857170761504, 4.61456163271528290142359440767, 5.46650567771203980404576724192, 7.59605441235377042188512780923, 8.656974229017541244274087158856, 9.076348820418074761085830970459, 10.87973145948670819238256605850, 11.40766600375226625905742618475, 12.03257226115836511858336278874
