| L(s) = 1 | + (−0.939 − 0.342i)2-s + (−0.173 − 0.984i)3-s + (0.766 + 0.642i)4-s + (0.766 − 0.642i)5-s + (−0.173 + 0.984i)6-s + (−0.897 + 1.55i)7-s + (−0.500 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.939 + 0.342i)10-s + (3.23 + 5.59i)11-s + (0.5 − 0.866i)12-s + (0.680 − 3.86i)13-s + (1.37 − 1.15i)14-s + (−0.766 − 0.642i)15-s + (0.173 + 0.984i)16-s + (4.33 + 1.57i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (−0.100 − 0.568i)3-s + (0.383 + 0.321i)4-s + (0.342 − 0.287i)5-s + (−0.0708 + 0.402i)6-s + (−0.339 + 0.587i)7-s + (−0.176 − 0.306i)8-s + (−0.313 + 0.114i)9-s + (−0.297 + 0.108i)10-s + (0.974 + 1.68i)11-s + (0.144 − 0.249i)12-s + (0.188 − 1.07i)13-s + (0.367 − 0.308i)14-s + (−0.197 − 0.165i)15-s + (0.0434 + 0.246i)16-s + (1.05 + 0.382i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11129 - 0.0922693i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11129 - 0.0922693i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 3 | \( 1 + (0.173 + 0.984i)T \) |
| 5 | \( 1 + (-0.766 + 0.642i)T \) |
| 19 | \( 1 + (1.99 - 3.87i)T \) |
| good | 7 | \( 1 + (0.897 - 1.55i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.23 - 5.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.680 + 3.86i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-4.33 - 1.57i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-2.62 - 2.20i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.70 + 0.619i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.84 + 4.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.08T + 37T^{2} \) |
| 41 | \( 1 + (-0.0777 - 0.440i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-6.73 + 5.65i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-2.21 + 0.805i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-1.36 - 1.14i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-7.80 - 2.84i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (2.49 + 2.09i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-11.5 + 4.18i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (1.98 - 1.66i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (0.341 + 1.93i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.334 - 1.89i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-2.26 + 3.92i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.466 - 2.64i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (17.1 + 6.25i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.46511427399136693940950026612, −9.834116417697592751317784457196, −9.042599691444027204792930076498, −8.084509856890895185886841472847, −7.27036539278113812506400432437, −6.26746004841224348071514643147, −5.39094111485914573254252569843, −3.83947570328947518378394905424, −2.40202589357730734245694723014, −1.28557012657556015397630664607,
0.978754286605511828910818757964, 2.93870289581108603866926750779, 3.99564894647901501748773314466, 5.37883690659445429232274825145, 6.45754965593183664463496736911, 6.94896575838087190878941459332, 8.420762177581336092441640008390, 9.028629244599460910857590887156, 9.798476321103840564583550354363, 10.74389397129095831531266442938
