| L(s) = 1 | + (−0.984 + 0.173i)2-s + (0.766 − 0.642i)3-s + (0.939 − 0.342i)4-s + (−1.27 + 3.50i)5-s + (−0.642 + 0.766i)6-s + (2.30 + 1.29i)7-s + (−0.866 + 0.5i)8-s + (0.173 − 0.984i)9-s + (0.646 − 3.66i)10-s + (2.37 + 4.11i)11-s + (0.499 − 0.866i)12-s + (−3.53 − 2.96i)13-s + (−2.49 − 0.869i)14-s + (1.27 + 3.50i)15-s + (0.766 − 0.642i)16-s + (−3.08 + 0.544i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 + 0.122i)2-s + (0.442 − 0.371i)3-s + (0.469 − 0.171i)4-s + (−0.569 + 1.56i)5-s + (−0.262 + 0.312i)6-s + (0.872 + 0.487i)7-s + (−0.306 + 0.176i)8-s + (0.0578 − 0.328i)9-s + (0.204 − 1.16i)10-s + (0.716 + 1.24i)11-s + (0.144 − 0.249i)12-s + (−0.979 − 0.821i)13-s + (−0.667 − 0.232i)14-s + (0.328 + 0.903i)15-s + (0.191 − 0.160i)16-s + (−0.749 + 0.132i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.375 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.592457 + 0.879102i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.592457 + 0.879102i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.984 - 0.173i)T \) |
| 3 | \( 1 + (-0.766 + 0.642i)T \) |
| 7 | \( 1 + (-2.30 - 1.29i)T \) |
| 19 | \( 1 + (-2.17 - 3.77i)T \) |
| good | 5 | \( 1 + (1.27 - 3.50i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (-2.37 - 4.11i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.53 + 2.96i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (3.08 - 0.544i)T + (15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-1.61 + 0.587i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (2.67 + 0.471i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (3.45 - 5.98i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 10.4iT - 37T^{2} \) |
| 41 | \( 1 + (8.20 - 6.88i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (4.95 + 1.80i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-10.1 - 1.78i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-1.38 - 3.79i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (1.26 + 7.20i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-4.38 - 12.0i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.683 - 0.120i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-0.584 + 1.60i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-8.46 - 10.0i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-5.70 - 6.79i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-7.25 - 4.18i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (9.68 + 8.12i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-0.0429 - 0.243i)T + (-91.1 + 33.1i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.47141252677980993921849204192, −9.715722791981016902574440702868, −8.744813002453312257568269066725, −7.75199965632356323031382073734, −7.29457504764612254026234302702, −6.65409195592601555593518522529, −5.35258062616413371099031289007, −3.92585969830086452783811907225, −2.71703199425502419343117420038, −1.85259315733396629467315757042,
0.62867713476877893585288164848, 1.90478736343864925590473819604, 3.60185680238688555012375095495, 4.54025661950977129692545785556, 5.24916457618833970739147529822, 6.84921992385718603493264716702, 7.75710232501890981664070110852, 8.557162061231227213988746522762, 9.001121631029237857941402067770, 9.664899496804709881774469121974
