| L(s) = 1 | + (−0.997 − 0.0713i)2-s + (−0.212 − 0.977i)3-s + (0.989 + 0.142i)4-s + (1.98 − 1.03i)5-s + (0.142 + 0.989i)6-s + (−0.653 − 1.19i)7-s + (−0.977 − 0.212i)8-s + (−0.909 + 0.415i)9-s + (−2.05 + 0.888i)10-s + (−1.60 − 1.38i)11-s + (−0.0713 − 0.997i)12-s + (−4.94 − 2.70i)13-s + (0.566 + 1.24i)14-s + (−1.43 − 1.71i)15-s + (0.959 + 0.281i)16-s + (−1.26 + 1.68i)17-s + ⋯ |
| L(s) = 1 | + (−0.705 − 0.0504i)2-s + (−0.122 − 0.564i)3-s + (0.494 + 0.0711i)4-s + (0.886 − 0.462i)5-s + (0.0580 + 0.404i)6-s + (−0.247 − 0.452i)7-s + (−0.345 − 0.0751i)8-s + (−0.303 + 0.138i)9-s + (−0.648 + 0.281i)10-s + (−0.482 − 0.418i)11-s + (−0.0205 − 0.287i)12-s + (−1.37 − 0.749i)13-s + (0.151 + 0.331i)14-s + (−0.369 − 0.443i)15-s + (0.239 + 0.0704i)16-s + (−0.306 + 0.409i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 + 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.144548 - 0.681320i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.144548 - 0.681320i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.997 + 0.0713i)T \) |
| 3 | \( 1 + (0.212 + 0.977i)T \) |
| 5 | \( 1 + (-1.98 + 1.03i)T \) |
| 23 | \( 1 + (4.28 - 2.15i)T \) |
| good | 7 | \( 1 + (0.653 + 1.19i)T + (-3.78 + 5.88i)T^{2} \) |
| 11 | \( 1 + (1.60 + 1.38i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (4.94 + 2.70i)T + (7.02 + 10.9i)T^{2} \) |
| 17 | \( 1 + (1.26 - 1.68i)T + (-4.78 - 16.3i)T^{2} \) |
| 19 | \( 1 + (-0.299 + 2.08i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-5.11 + 0.735i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (0.875 - 0.562i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-3.05 + 8.19i)T + (-27.9 - 24.2i)T^{2} \) |
| 41 | \( 1 + (3.31 - 7.25i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (6.00 - 1.30i)T + (39.1 - 17.8i)T^{2} \) |
| 47 | \( 1 + (1.39 - 1.39i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.68 - 1.46i)T + (28.6 - 44.5i)T^{2} \) |
| 59 | \( 1 + (0.317 + 1.08i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-0.435 - 0.678i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-1.14 + 16.0i)T + (-66.3 - 9.53i)T^{2} \) |
| 71 | \( 1 + (7.09 + 8.18i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.494 + 0.369i)T + (20.5 - 70.0i)T^{2} \) |
| 79 | \( 1 + (13.0 - 3.84i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (1.39 + 0.520i)T + (62.7 + 54.3i)T^{2} \) |
| 89 | \( 1 + (-14.7 - 9.49i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-10.5 + 3.94i)T + (73.3 - 63.5i)T^{2} \) |
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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
−10.06406949380732608320087267821, −9.336655751931082835483434910613, −8.293656104638462488050320437033, −7.61471336733893249404244205165, −6.62243411553746017999112485609, −5.77405223395717800746369479082, −4.78340617431344179317987548390, −2.98846487460558581180279077916, −1.92962849097815613111879860633, −0.43111146209327212977559827696,
2.05441826084251451193226278690, 2.89198349835720103099075672760, 4.57389116021497138004022510406, 5.53904710671726254056882324191, 6.51680242636327819080643872899, 7.27437628022721341351995953970, 8.474043174697759899409085132280, 9.339375447805332171580879146166, 10.08218996249561124476520724339, 10.30736338258288087754519741339
