L(s) = 1 | + 2.37i·2-s + (−2.91 − 0.699i)3-s − 1.64·4-s + 3.76i·5-s + (1.66 − 6.93i)6-s + 9.72·7-s + 5.58i·8-s + (8.02 + 4.08i)9-s − 8.95·10-s + 6.42i·11-s + (4.80 + 1.15i)12-s − 19.9·13-s + 23.1i·14-s + (2.63 − 10.9i)15-s − 19.8·16-s − 12.2i·17-s + ⋯ |
L(s) = 1 | + 1.18i·2-s + (−0.972 − 0.233i)3-s − 0.412·4-s + 0.753i·5-s + (0.277 − 1.15i)6-s + 1.38·7-s + 0.698i·8-s + (0.891 + 0.453i)9-s − 0.895·10-s + 0.584i·11-s + (0.400 + 0.0961i)12-s − 1.53·13-s + 1.65i·14-s + (0.175 − 0.732i)15-s − 1.24·16-s − 0.718i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.233i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.131448 + 1.11153i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.131448 + 1.11153i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.91 + 0.699i)T \) |
| 67 | \( 1 - 8.18T \) |
good | 2 | \( 1 - 2.37iT - 4T^{2} \) |
| 5 | \( 1 - 3.76iT - 25T^{2} \) |
| 7 | \( 1 - 9.72T + 49T^{2} \) |
| 11 | \( 1 - 6.42iT - 121T^{2} \) |
| 13 | \( 1 + 19.9T + 169T^{2} \) |
| 17 | \( 1 + 12.2iT - 289T^{2} \) |
| 19 | \( 1 + 16.2T + 361T^{2} \) |
| 23 | \( 1 - 29.8iT - 529T^{2} \) |
| 29 | \( 1 - 38.6iT - 841T^{2} \) |
| 31 | \( 1 - 11.7T + 961T^{2} \) |
| 37 | \( 1 - 14.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 6.17iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 11.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 1.34iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 74.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 58.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 36.2T + 3.72e3T^{2} \) |
| 71 | \( 1 - 128. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 105.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 117.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 56.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 71.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 118.T + 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.55908472289033559615642851145, −11.53016163845546945367870713140, −10.98122404393897767535524773317, −9.773191313222462900184914326205, −8.139106298363207717251697829886, −7.24774278697650721746933576152, −6.78143543509393362495527805102, −5.29813398761494378659409152407, −4.78937533058225514108238073104, −2.10687321462990366764024242198,
0.72093869172424618446988044954, 2.16885704166747345578701868799, 4.28023312215724257629358688623, 4.90681153578782208601717949089, 6.35864188764146663282172775581, 7.84967006158597810101528936165, 9.073440539438911023002680115401, 10.29694309782367579580576649211, 10.84487824407582219536286882992, 11.83316297869399198831316034813