L(s) = 1 | + (0.596 + 0.596i)5-s + 29.0i·7-s + (12.1 + 12.1i)11-s + (−48.5 + 48.5i)13-s − 86.7·17-s + (54.8 − 54.8i)19-s − 70.2i·23-s − 124. i·25-s + (−63.4 + 63.4i)29-s + 8.86·31-s + (−17.3 + 17.3i)35-s + (−21.7 − 21.7i)37-s − 153. i·41-s + (120. + 120. i)43-s − 99.9·47-s + ⋯ |
L(s) = 1 | + (0.0533 + 0.0533i)5-s + 1.57i·7-s + (0.332 + 0.332i)11-s + (−1.03 + 1.03i)13-s − 1.23·17-s + (0.662 − 0.662i)19-s − 0.636i·23-s − 0.994i·25-s + (−0.405 + 0.405i)29-s + 0.0513·31-s + (−0.0838 + 0.0838i)35-s + (−0.0964 − 0.0964i)37-s − 0.583i·41-s + (0.428 + 0.428i)43-s − 0.310·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.206i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.4733413119\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4733413119\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.596 - 0.596i)T + 125iT^{2} \) |
| 7 | \( 1 - 29.0iT - 343T^{2} \) |
| 11 | \( 1 + (-12.1 - 12.1i)T + 1.33e3iT^{2} \) |
| 13 | \( 1 + (48.5 - 48.5i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + 86.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + (-54.8 + 54.8i)T - 6.85e3iT^{2} \) |
| 23 | \( 1 + 70.2iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (63.4 - 63.4i)T - 2.43e4iT^{2} \) |
| 31 | \( 1 - 8.86T + 2.97e4T^{2} \) |
| 37 | \( 1 + (21.7 + 21.7i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + 153. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-120. - 120. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + 99.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + (389. + 389. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + (324. + 324. i)T + 2.05e5iT^{2} \) |
| 61 | \( 1 + (0.339 - 0.339i)T - 2.26e5iT^{2} \) |
| 67 | \( 1 + (565. - 565. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 - 419. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 374. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 705.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (947. - 947. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + 4.72iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 379.T + 9.12e5T^{2} \) |
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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
−10.90647338199989488279847623323, −9.593795485341615356465971292085, −9.165027208085639894638599176703, −8.328668969449671544020612500630, −7.03653788826206833076382255851, −6.34738137776636664925651830341, −5.17564271712802109514930116070, −4.38004643541493700768853934196, −2.71544365626811988175121895301, −1.98300546976644516659445085855,
0.13581468507461638303729159335, 1.41451703920941481485613579523, 3.08522178207019048314004738374, 4.08839724576060602896219454831, 5.09308270395957164399455537116, 6.25375197517412538361876269401, 7.39702860419189014664190604308, 7.74513683163713217631685652672, 9.114349158646396303980213202687, 9.929774677559431678165602365786