| L(s) = 1 | + (1.36 − 1.06i)3-s + (2.18 − 1.82i)5-s + (−0.145 + 0.0527i)7-s + (0.715 − 2.91i)9-s + (−4.10 − 3.44i)11-s + (0.00459 + 0.0260i)13-s + (1.01 − 4.82i)15-s + (2.88 + 4.98i)17-s + (−3.10 + 5.37i)19-s + (−0.141 + 0.226i)21-s + (3.35 + 1.22i)23-s + (0.539 − 3.05i)25-s + (−2.13 − 4.73i)27-s + (0.0327 − 0.185i)29-s + (2.56 + 0.932i)31-s + ⋯ |
| L(s) = 1 | + (0.786 − 0.617i)3-s + (0.975 − 0.818i)5-s + (−0.0548 + 0.0199i)7-s + (0.238 − 0.971i)9-s + (−1.23 − 1.03i)11-s + (0.00127 + 0.00723i)13-s + (0.262 − 1.24i)15-s + (0.698 + 1.20i)17-s + (−0.711 + 1.23i)19-s + (−0.0308 + 0.0495i)21-s + (0.700 + 0.254i)23-s + (0.107 − 0.611i)25-s + (−0.411 − 0.911i)27-s + (0.00609 − 0.0345i)29-s + (0.460 + 0.167i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.357 + 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.357 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.61060 - 1.10768i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61060 - 1.10768i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.36 + 1.06i)T \) |
| good | 5 | \( 1 + (-2.18 + 1.82i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (0.145 - 0.0527i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (4.10 + 3.44i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.00459 - 0.0260i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-2.88 - 4.98i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.10 - 5.37i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.35 - 1.22i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.0327 + 0.185i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-2.56 - 0.932i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (4.11 + 7.13i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.00921 + 0.0522i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-6.21 - 5.21i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-11.8 + 4.31i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + (3.24 - 2.72i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (4.47 - 1.62i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.03 + 5.88i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-1.35 - 2.35i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.37 - 12.7i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.880 - 4.99i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.538 + 3.05i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-2.11 + 3.66i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (9.25 + 7.76i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.75869042103341875430040291250, −10.02308120312059993665612933341, −8.928572580222432387630074012448, −8.377206190019837603550394621561, −7.51071985317335050215094324198, −6.03722271203278386734717142533, −5.54110094728698716383875046904, −3.86450391698102670203612719089, −2.55838044532468639950439194047, −1.29996393186609573933498452627,
2.35398462481250048045179310712, 2.89705182185519106285252726166, 4.58046644869176262251062010663, 5.40867748373970809384436299910, 6.87012044204002878563079583665, 7.56189278701211284061306429949, 8.812419114998765375921790731781, 9.663845489192831361601167250568, 10.31592365997297766520406393646, 10.88218130632521320846345994501
