| L(s) = 1 | + (1.41 + 0.0781i)2-s + (0.889 + 1.48i)3-s + (1.98 + 0.220i)4-s + (−0.479 − 0.174i)5-s + (1.13 + 2.16i)6-s + (−1.63 − 0.287i)7-s + (2.78 + 0.467i)8-s + (−1.41 + 2.64i)9-s + (−0.663 − 0.283i)10-s + (−0.576 − 1.58i)11-s + (1.43 + 3.15i)12-s + (−3.81 − 4.54i)13-s + (−2.28 − 0.533i)14-s + (−0.167 − 0.867i)15-s + (3.90 + 0.877i)16-s + (5.08 − 2.93i)17-s + ⋯ |
| L(s) = 1 | + (0.998 + 0.0552i)2-s + (0.513 + 0.858i)3-s + (0.993 + 0.110i)4-s + (−0.214 − 0.0780i)5-s + (0.465 + 0.885i)6-s + (−0.616 − 0.108i)7-s + (0.986 + 0.165i)8-s + (−0.472 + 0.881i)9-s + (−0.209 − 0.0897i)10-s + (−0.173 − 0.477i)11-s + (0.415 + 0.909i)12-s + (−1.05 − 1.26i)13-s + (−0.609 − 0.142i)14-s + (−0.0431 − 0.224i)15-s + (0.975 + 0.219i)16-s + (1.23 − 0.712i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 - 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.767 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.10693 + 0.763775i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.10693 + 0.763775i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 - 0.0781i)T \) |
| 3 | \( 1 + (-0.889 - 1.48i)T \) |
| good | 5 | \( 1 + (0.479 + 0.174i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (1.63 + 0.287i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (0.576 + 1.58i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (3.81 + 4.54i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-5.08 + 2.93i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.91 - 3.32i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.765 - 4.34i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.748 - 0.628i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-6.88 + 1.21i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (5.88 - 3.39i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.01 + 2.40i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.826 + 0.300i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.0117 - 0.0666i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 5.83T + 53T^{2} \) |
| 59 | \( 1 + (0.521 - 1.43i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (10.6 + 1.88i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (7.05 - 5.92i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-8.21 - 14.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.650 + 1.12i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.66 - 4.36i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (1.44 - 1.72i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-11.2 - 6.52i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.9 + 5.06i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.47552713755985449110059048715, −11.66507416026079428073011927042, −10.29558212264119877972242178784, −9.934155290883120119004903927650, −8.239232071865470034350313618859, −7.45388186675005841444266601931, −5.84018912576405051237118982259, −4.98062888755672742200776681435, −3.62214882621281964237250715235, −2.80849252664476032013911008732,
2.05016773101652453524154570506, 3.26887029921371164850952065354, 4.64413748840113810244200681300, 6.18482059876657109059854934426, 6.97380108200417679928179109408, 7.84980488913155394065289929457, 9.276671052192514719881747556617, 10.39282952146997987297398711289, 11.85886426693303620265694293715, 12.25885958313103040287995692335
