| L(s) = 1 | + (0.173 + 0.984i)2-s + (−1.93 + 1.61i)3-s + (−0.939 + 0.342i)4-s + (2.31 + 0.841i)5-s + (−1.93 − 1.61i)6-s + (1.39 − 2.41i)7-s + (−0.5 − 0.866i)8-s + (0.581 − 3.30i)9-s + (−0.427 + 2.42i)10-s + (0.839 + 1.45i)11-s + (1.26 − 2.18i)12-s + (4.86 + 4.07i)13-s + (2.62 + 0.955i)14-s + (−5.82 + 2.12i)15-s + (0.766 − 0.642i)16-s + (0.862 + 4.89i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (−1.11 + 0.935i)3-s + (−0.469 + 0.171i)4-s + (1.03 + 0.376i)5-s + (−0.788 − 0.661i)6-s + (0.527 − 0.914i)7-s + (−0.176 − 0.306i)8-s + (0.193 − 1.10i)9-s + (−0.135 + 0.765i)10-s + (0.253 + 0.438i)11-s + (0.363 − 0.630i)12-s + (1.34 + 1.13i)13-s + (0.701 + 0.255i)14-s + (−1.50 + 0.547i)15-s + (0.191 − 0.160i)16-s + (0.209 + 1.18i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 - 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.492411 + 1.28059i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.492411 + 1.28059i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + (1.93 - 1.61i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (-2.31 - 0.841i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.39 + 2.41i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.839 - 1.45i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.86 - 4.07i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.862 - 4.89i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-2.34 + 0.854i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.03 + 5.84i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (3.64 - 6.30i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.550T + 37T^{2} \) |
| 41 | \( 1 + (1.99 - 1.67i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (2.69 + 0.982i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.129 + 0.734i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-1.38 + 0.502i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (0.861 + 4.88i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (8.77 - 3.19i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-2.00 + 11.3i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-6.57 - 2.39i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (4.73 - 3.97i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (4.52 - 3.79i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (7.58 - 13.1i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.28 - 4.43i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-2.49 - 14.1i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.68650459236010255312537345434, −10.00807671100567519260410701769, −9.192737974885692140327467019810, −8.133297313398493787650574887269, −6.78340923333800383264516031453, −6.29362350008280541281484813876, −5.46309274053723894194462667195, −4.45265341686963539584475055913, −3.81465345651388496250380971085, −1.54671225328993088367196161702,
0.901959137249014442573858037422, 1.82742095669616719810602804685, 3.17368033104422706311155761592, 5.00132730941968997850417812492, 5.67727712892762946170010564665, 6.05889755972971139845345638908, 7.37772978054034809148578301160, 8.580121395655914267028913863394, 9.212028708624979128153766926755, 10.33243789593829133381546854826
