| L(s) = 1 | + (0.236 + 1.39i)2-s + (0.0980 − 0.995i)3-s + (−1.88 + 0.659i)4-s + (−1.27 + 0.682i)5-s + (1.41 − 0.0985i)6-s + (−0.687 − 3.45i)7-s + (−1.36 − 2.47i)8-s + (−0.980 − 0.195i)9-s + (−1.25 − 1.61i)10-s + (4.18 − 5.10i)11-s + (0.470 + 1.94i)12-s + (−1.37 + 2.57i)13-s + (4.65 − 1.77i)14-s + (0.554 + 1.33i)15-s + (3.13 − 2.48i)16-s + (2.00 − 4.82i)17-s + ⋯ |
| L(s) = 1 | + (0.167 + 0.985i)2-s + (0.0565 − 0.574i)3-s + (−0.944 + 0.329i)4-s + (−0.570 + 0.305i)5-s + (0.575 − 0.0402i)6-s + (−0.260 − 1.30i)7-s + (−0.482 − 0.875i)8-s + (−0.326 − 0.0650i)9-s + (−0.396 − 0.511i)10-s + (1.26 − 1.53i)11-s + (0.135 + 0.561i)12-s + (−0.382 + 0.715i)13-s + (1.24 − 0.474i)14-s + (0.143 + 0.345i)15-s + (0.782 − 0.622i)16-s + (0.485 − 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.956396 - 0.370263i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.956396 - 0.370263i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.236 - 1.39i)T \) |
| 3 | \( 1 + (-0.0980 + 0.995i)T \) |
| good | 5 | \( 1 + (1.27 - 0.682i)T + (2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (0.687 + 3.45i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-4.18 + 5.10i)T + (-2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (1.37 - 2.57i)T + (-7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-2.00 + 4.82i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.106 + 0.349i)T + (-15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (4.11 - 2.74i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (-5.85 + 4.80i)T + (5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (-2.89 + 2.89i)T - 31iT^{2} \) |
| 37 | \( 1 + (10.8 + 3.30i)T + (30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (5.71 + 8.55i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-0.299 - 3.04i)T + (-42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (0.979 + 0.405i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-9.08 - 7.45i)T + (10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (0.941 + 1.76i)T + (-32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (-6.75 - 0.665i)T + (59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-7.01 - 0.690i)T + (65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (7.00 - 1.39i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (2.63 - 13.2i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (1.48 - 0.614i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-11.6 + 3.51i)T + (69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (7.92 + 5.29i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (0.437 - 0.437i)T - 97iT^{2} \) |
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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
−11.58048996831243914450259000339, −10.16980170346908761892246653611, −9.138830164743708265474544771003, −8.189900090465953674632867359828, −7.21303063521076267908608777481, −6.78833273527382285512667404185, −5.68833036089361802679505852514, −4.13280284968384913133156340025, −3.43333550983469229091868823500, −0.67710908277381709462248197385,
1.90741317549485169462721688809, 3.30815444904426065257477388634, 4.35842397811882763088693010239, 5.23264744639964654704063650291, 6.47640345149573296074352643501, 8.280302265853250164035276745058, 8.780824862959712556634529740372, 9.941085337348578201395790682007, 10.26868532760671083561576766722, 11.82797866701111521268545151162
