| L(s) = 1 | + (−0.908 − 1.08i)2-s + (0.654 + 0.755i)3-s + (−0.350 + 1.96i)4-s + (0.0443 + 0.308i)5-s + (0.224 − 1.39i)6-s + (0.778 − 1.70i)7-s + (2.45 − 1.40i)8-s + (−0.142 + 0.989i)9-s + (0.293 − 0.328i)10-s + (−0.132 + 0.450i)11-s + (−1.71 + 1.02i)12-s + (4.84 − 2.21i)13-s + (−2.55 + 0.704i)14-s + (−0.203 + 0.235i)15-s + (−3.75 − 1.37i)16-s + (−3.35 + 5.21i)17-s + ⋯ |
| L(s) = 1 | + (−0.642 − 0.766i)2-s + (0.378 + 0.436i)3-s + (−0.175 + 0.984i)4-s + (0.0198 + 0.137i)5-s + (0.0916 − 0.570i)6-s + (0.294 − 0.644i)7-s + (0.867 − 0.498i)8-s + (−0.0474 + 0.329i)9-s + (0.0929 − 0.103i)10-s + (−0.0398 + 0.135i)11-s + (−0.495 + 0.295i)12-s + (1.34 − 0.613i)13-s + (−0.683 + 0.188i)14-s + (−0.0526 + 0.0607i)15-s + (−0.938 − 0.344i)16-s + (−0.812 + 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.25673 - 0.215968i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25673 - 0.215968i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.908 + 1.08i)T \) |
| 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (-4.73 + 0.741i)T \) |
| good | 5 | \( 1 + (-0.0443 - 0.308i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (-0.778 + 1.70i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (0.132 - 0.450i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-4.84 + 2.21i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (3.35 - 5.21i)T + (-7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (0.457 + 0.712i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-0.704 + 1.09i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-4.98 - 4.32i)T + (4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.124 + 0.866i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-1.07 - 7.47i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-7.10 + 6.15i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 2.81iT - 47T^{2} \) |
| 53 | \( 1 + (-1.73 + 3.79i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (2.13 + 4.66i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (8.17 - 9.43i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-1.44 - 4.91i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (2.07 + 7.06i)T + (-59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-4.75 + 3.05i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (5.11 + 11.1i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (9.65 + 1.38i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (0.723 - 0.626i)T + (12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (9.74 - 1.40i)T + (93.0 - 27.3i)T^{2} \) |
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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
−10.77131066777147664186338748166, −10.09505933404068866322592798330, −8.808531154391176768010242068731, −8.499148305829180694114319489587, −7.43533743567948057637892293805, −6.33945810141394209382041169946, −4.67628548864253876397136161496, −3.82283409269540891868101750976, −2.77905054014856088625024171427, −1.23922027060895278174284993357,
1.17278524830519905555751248434, 2.61728330842599152483688101275, 4.40705146902598930066795539908, 5.52408819506509396116579484468, 6.48459597891119980926553642029, 7.24074056848040041138528086553, 8.327396700386106062034510468040, 8.920942071192123140912637189811, 9.476193704907546701062811909385, 10.91375061386085698478323021391
