L(s) = 1 | − 24.8·3-s + (−53.4 − 16.3i)5-s + 100. i·7-s + 372.·9-s + 5.36i·11-s − 506.·13-s + (1.32e3 + 406. i)15-s − 1.45e3i·17-s − 722. i·19-s − 2.48e3i·21-s − 3.08e3i·23-s + (2.58e3 + 1.75e3i)25-s − 3.20e3·27-s − 7.29e3i·29-s + 8.60e3·31-s + ⋯ |
L(s) = 1 | − 1.59·3-s + (−0.956 − 0.293i)5-s + 0.773i·7-s + 1.53·9-s + 0.0133i·11-s − 0.831·13-s + (1.52 + 0.466i)15-s − 1.22i·17-s − 0.459i·19-s − 1.23i·21-s − 1.21i·23-s + (0.828 + 0.560i)25-s − 0.846·27-s − 1.61i·29-s + 1.60·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.468 - 0.883i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.1237983716\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1237983716\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (53.4 + 16.3i)T \) |
good | 3 | \( 1 + 24.8T + 243T^{2} \) |
| 7 | \( 1 - 100. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 5.36iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 506.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.45e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 722. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 3.08e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 7.29e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 8.60e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.27e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.81e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.34e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.21e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.33e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.09e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 1.09e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 1.30e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.60e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.42e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 4.86e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.97e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.76e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.68e4iT - 8.58e9T^{2} \) |
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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
−11.37431827168702764886775391219, −10.40006389411905055132998259688, −9.353313142863691093461894688761, −8.158518383974453480544818597533, −7.07408382265674378524176493490, −6.18326149709569437957212798059, −5.00407199567553602199200168341, −4.52207047448248497522986586226, −2.67653843768630735161646246310, −0.72437881763799296003080238129,
0.06849415050395067155260929386, 1.32179712212076168431779208156, 3.50191521598411221876910059153, 4.52335133707338498172129165606, 5.48735949391952864955555040065, 6.69196262314558276163998445772, 7.29006697548640502337281661458, 8.409813396203890540131839679717, 10.13002247785352295130761759301, 10.53513704992616466680054098889