| L(s) = 1 | + (0.654 + 0.755i)2-s − 3-s + (−0.142 + 0.989i)4-s + (2.43 + 1.56i)5-s + (−0.654 − 0.755i)6-s + (−0.349 + 0.102i)7-s + (−0.841 + 0.540i)8-s + 9-s + (0.412 + 2.86i)10-s + (3.15 + 1.01i)11-s + (0.142 − 0.989i)12-s + (0.466 − 3.24i)13-s + (−0.306 − 0.196i)14-s + (−2.43 − 1.56i)15-s + (−0.959 − 0.281i)16-s + (2.97 + 6.50i)17-s + ⋯ |
| L(s) = 1 | + (0.463 + 0.534i)2-s − 0.577·3-s + (−0.0711 + 0.494i)4-s + (1.09 + 0.701i)5-s + (−0.267 − 0.308i)6-s + (−0.132 + 0.0387i)7-s + (−0.297 + 0.191i)8-s + 0.333·9-s + (0.130 + 0.907i)10-s + (0.951 + 0.306i)11-s + (0.0410 − 0.285i)12-s + (0.129 − 0.900i)13-s + (−0.0819 − 0.0526i)14-s + (−0.629 − 0.404i)15-s + (−0.239 − 0.0704i)16-s + (0.720 + 1.57i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0645 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0645 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.32989 + 1.41869i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32989 + 1.41869i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.654 - 0.755i)T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + (-3.15 - 1.01i)T \) |
| good | 5 | \( 1 + (-2.43 - 1.56i)T + (2.07 + 4.54i)T^{2} \) |
| 7 | \( 1 + (0.349 - 0.102i)T + (5.88 - 3.78i)T^{2} \) |
| 13 | \( 1 + (-0.466 + 3.24i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-2.97 - 6.50i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.0458 + 0.100i)T + (-12.4 - 14.3i)T^{2} \) |
| 23 | \( 1 + (4.13 + 1.21i)T + (19.3 + 12.4i)T^{2} \) |
| 29 | \( 1 + (1.64 - 3.59i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.311 + 2.16i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-1.20 - 8.41i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (1.50 + 1.73i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-4.80 + 3.09i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (1.49 - 1.72i)T + (-6.68 - 46.5i)T^{2} \) |
| 53 | \( 1 + (-11.5 + 3.39i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-1.47 + 1.70i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (6.13 - 7.08i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (0.843 + 0.973i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (0.238 - 0.522i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (2.96 + 0.871i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (14.2 + 9.14i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-6.51 + 1.91i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (2.54 + 5.57i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-15.2 + 9.82i)T + (40.2 - 88.2i)T^{2} \) |
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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
−10.35443061122936173366098606877, −10.08767801003509171162690716603, −8.887497715656203459201763559314, −7.83137405738994623373726283089, −6.79680323656710900114425243523, −6.04629288729705753752956340779, −5.65920789285238524513282403342, −4.29365971724813225618316340127, −3.20465509274608860263064554754, −1.71102258032039878124732729711,
1.03518739011285707227581938792, 2.17606629638161807192622328028, 3.73808781025877358475560410851, 4.76646034092497137651381185320, 5.64072613916838064669238358185, 6.28773874528023719851494008817, 7.34715474779520382130213103552, 8.899365062515479675444411569745, 9.504726771367228216406519297642, 10.05375071858841046728852448556
