L(s) = 1 | + (−1.5 + 0.866i)2-s + (−1.5 + 0.866i)3-s + (0.5 − 0.866i)4-s + (−1.5 + 0.866i)5-s + (1.5 − 2.59i)6-s − 1.73i·7-s − 1.73i·8-s + (1.5 − 2.59i)9-s + (1.5 − 2.59i)10-s + (−3 + 1.73i)11-s + 1.73i·12-s + (−2.5 − 2.59i)13-s + (1.49 + 2.59i)14-s + (1.5 − 2.59i)15-s + (2.49 + 4.33i)16-s + (−1.5 − 2.59i)17-s + ⋯ |
L(s) = 1 | + (−1.06 + 0.612i)2-s + (−0.866 + 0.499i)3-s + (0.250 − 0.433i)4-s + (−0.670 + 0.387i)5-s + (0.612 − 1.06i)6-s − 0.654i·7-s − 0.612i·8-s + (0.5 − 0.866i)9-s + (0.474 − 0.821i)10-s + (−0.904 + 0.522i)11-s + 0.500i·12-s + (−0.693 − 0.720i)13-s + (0.400 + 0.694i)14-s + (0.387 − 0.670i)15-s + (0.624 + 1.08i)16-s + (−0.363 − 0.630i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.428 + 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.428 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.5 - 0.866i)T \) |
| 13 | \( 1 + (2.5 + 2.59i)T \) |
good | 2 | \( 1 + (1.5 - 0.866i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 1.73iT - 7T^{2} \) |
| 11 | \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 - 0.866i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (7.5 - 4.33i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.5 + 2.59i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 12.1iT - 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + (4.5 + 2.59i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (-3 - 1.73i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + 5T + 61T^{2} \) |
| 67 | \( 1 - 12.1iT - 67T^{2} \) |
| 71 | \( 1 + (7.5 - 4.33i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.5 + 2.59i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-13.5 - 7.79i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 15.5iT - 97T^{2} \) |
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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
−12.98354886104095708706834147321, −12.03435267932437668437033047366, −10.56959718220958168531791237660, −10.27377683308262605034854737643, −8.923219890284389932905477106741, −7.47458363107256614223360335633, −7.03439552964069623066767208973, −5.29878391325248697301060944214, −3.79158084802677213002512374174, 0,
2.15276262389819621973419206531, 4.75082974210359934621131928119, 6.05908241654551215166689785058, 7.72713435356032831754083300337, 8.480946020083334206735078351896, 9.780559707535456130152025476849, 10.84897546948938748494242978061, 11.66807984831276045420323313624, 12.35165226491345955772984980588