L(s) = 1 | + (−22.6 − 39.1i)5-s + 80.9·7-s + 74.1·11-s + (−249. − 144. i)13-s + (−255. − 441. i)17-s + (−356. + 58.3i)19-s + (122. − 212. i)23-s + (−709. + 1.22e3i)25-s + (932. + 538. i)29-s − 96.0i·31-s + (−1.82e3 − 3.16e3i)35-s + 1.33e3i·37-s + (1.75e3 − 1.01e3i)41-s + (−1.40e3 − 2.43e3i)43-s + (1.40e3 − 2.44e3i)47-s + ⋯ |
L(s) = 1 | + (−0.904 − 1.56i)5-s + 1.65·7-s + 0.612·11-s + (−1.47 − 0.853i)13-s + (−0.882 − 1.52i)17-s + (−0.986 + 0.161i)19-s + (0.231 − 0.400i)23-s + (−1.13 + 1.96i)25-s + (1.10 + 0.640i)29-s − 0.0999i·31-s + (−1.49 − 2.58i)35-s + 0.975i·37-s + (1.04 − 0.602i)41-s + (−0.761 − 1.31i)43-s + (0.637 − 1.10i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.8485091961\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8485091961\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (356. - 58.3i)T \) |
good | 5 | \( 1 + (22.6 + 39.1i)T + (-312.5 + 541. i)T^{2} \) |
| 7 | \( 1 - 80.9T + 2.40e3T^{2} \) |
| 11 | \( 1 - 74.1T + 1.46e4T^{2} \) |
| 13 | \( 1 + (249. + 144. i)T + (1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 + (255. + 441. i)T + (-4.17e4 + 7.23e4i)T^{2} \) |
| 23 | \( 1 + (-122. + 212. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-932. - 538. i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + 96.0iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.33e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-1.75e3 + 1.01e3i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (1.40e3 + 2.43e3i)T + (-1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-1.40e3 + 2.44e3i)T + (-2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + (-754. - 435. i)T + (3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (5.31e3 - 3.06e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (14.3 - 24.8i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-3.61e3 - 2.08e3i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (2.89e3 - 1.66e3i)T + (1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + (457. + 791. i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-413. + 238. i)T + (1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 - 1.08e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + (1.05e4 + 6.06e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + (1.04e4 - 6.04e3i)T + (4.42e7 - 7.66e7i)T^{2} \) |
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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
−9.024500568083640070821693581432, −8.594179739994513210138948045020, −7.79326871123091801449135357261, −7.04860910916220666389331654354, −5.26529126215809313278598644796, −4.81330407884503838418437683459, −4.19739361810559375051513177737, −2.42770953761033941844009203135, −1.13906812322841942305886087480, −0.21354275472937595978327057912,
1.76149185395201826630556078702, 2.65448771027785264559039275810, 4.17906357904254971299462314606, 4.51562556169540765334250187730, 6.21076190343604812754223060699, 6.93873183865972223131344351878, 7.78111529728106080074916394492, 8.384180710174193784623746150539, 9.603650883794172882392912139352, 10.83512250854923433566062032725