L(s) = 1 | + (0.5 − 0.866i)2-s + (1.5 − 0.866i)3-s + (0.500 + 0.866i)4-s − 1.73i·6-s + (−1.5 + 2.59i)7-s + 3·8-s + (1.5 − 2.59i)9-s + (1 − 1.73i)11-s + (1.5 + 0.866i)12-s + (−1 − 1.73i)13-s + (1.5 + 2.59i)14-s + (0.500 − 0.866i)16-s − 4·17-s + (−1.5 − 2.59i)18-s − 8·19-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.866 − 0.499i)3-s + (0.250 + 0.433i)4-s − 0.707i·6-s + (−0.566 + 0.981i)7-s + 1.06·8-s + (0.5 − 0.866i)9-s + (0.301 − 0.522i)11-s + (0.433 + 0.249i)12-s + (−0.277 − 0.480i)13-s + (0.400 + 0.694i)14-s + (0.125 − 0.216i)16-s − 0.970·17-s + (−0.353 − 0.612i)18-s − 1.83·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.81218 - 0.659580i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.81218 - 0.659580i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T + (-1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.5 + 6.06i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + (-7 - 12.1i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + (-3 + 5.19i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.32700965709325379584105972140, −11.45570073539518445627371394529, −10.30002848156502976792853398308, −8.981709833724890215635177267602, −8.373990530836657260994336089461, −7.13187191844188848339622197555, −6.11466619712105133099479900206, −4.24175257649284199512192035834, −3.03345464767430081642382528285, −2.14335186687034947077085323824,
2.15270273663897087433423311516, 4.03678102902236035474044043637, 4.73051919463241812527567815526, 6.51698324827261283119425524520, 7.07311713550832796347836079733, 8.338431592470254857090624503554, 9.499642426053140473710067247109, 10.35775039409588314325214837352, 11.04048345028113062837883033782, 12.79054183804456881792857621252