L(s) = 1 | + 4.96i·2-s − 8.66·4-s − 41.3·5-s + 1.08·7-s + 36.4i·8-s − 205. i·10-s − 142. i·11-s − 35.2i·13-s + 5.39i·14-s − 319.·16-s − 527.·17-s + 76.0·19-s + 358.·20-s + 708.·22-s + 138. i·23-s + ⋯ |
L(s) = 1 | + 1.24i·2-s − 0.541·4-s − 1.65·5-s + 0.0221·7-s + 0.569i·8-s − 2.05i·10-s − 1.17i·11-s − 0.208i·13-s + 0.0275i·14-s − 1.24·16-s − 1.82·17-s + 0.210·19-s + 0.897·20-s + 1.46·22-s + 0.262i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0950 - 0.995i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0950 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.041193801\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.041193801\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 59 | \( 1 + (330. - 3.46e3i)T \) |
good | 2 | \( 1 - 4.96iT - 16T^{2} \) |
| 5 | \( 1 + 41.3T + 625T^{2} \) |
| 7 | \( 1 - 1.08T + 2.40e3T^{2} \) |
| 11 | \( 1 + 142. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 35.2iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 527.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 76.0T + 1.30e5T^{2} \) |
| 23 | \( 1 - 138. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.04e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.36e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 2.07e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.60e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 746. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 613. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 3.30e3T + 7.89e6T^{2} \) |
| 61 | \( 1 - 3.02e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 3.67e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 1.82e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 2.69e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 3.08e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 7.95e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 5.14e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.30e4iT - 8.85e7T^{2} \) |
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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
−10.67659498485399410394987487360, −9.016914632853652672815988012272, −8.344895431255666587835778890396, −7.78947960916367021629753616128, −6.86619578189944252795825368346, −6.10860926669132118620474712262, −4.86539266936343995170832990379, −4.00979867837455175509203473646, −2.72942947521523359136221937524, −0.52891481922997491268863271095,
0.54839391185612415300572686677, 1.99136752446078903878055138148, 3.08899869873966879742008410400, 4.22219343308693966850527432903, 4.61940621563684592771913702793, 6.71190951830077900521879187534, 7.26476652159373395479634233668, 8.434755854366738069024821985969, 9.269797790394506940056683370696, 10.35483619249186139314391453717