L(s) = 1 | + (0.856 + 0.515i)2-s + (0.370 − 0.928i)3-s + (0.468 + 0.883i)4-s + (−1.41 − 0.153i)5-s + (0.796 − 0.605i)6-s + (2.98 + 1.00i)7-s + (−0.0541 + 0.998i)8-s + (−0.725 − 0.687i)9-s + (−1.13 − 0.860i)10-s + (2.18 + 3.23i)11-s + (0.994 − 0.108i)12-s + (3.44 − 3.26i)13-s + (2.03 + 2.39i)14-s + (−0.666 + 1.25i)15-s + (−0.561 + 0.827i)16-s + (1.15 − 0.388i)17-s + ⋯ |
L(s) = 1 | + (0.605 + 0.364i)2-s + (0.213 − 0.536i)3-s + (0.234 + 0.441i)4-s + (−0.632 − 0.0687i)5-s + (0.325 − 0.247i)6-s + (1.12 + 0.379i)7-s + (−0.0191 + 0.353i)8-s + (−0.241 − 0.229i)9-s + (−0.358 − 0.272i)10-s + (0.660 + 0.973i)11-s + (0.286 − 0.0312i)12-s + (0.956 − 0.906i)13-s + (0.544 + 0.640i)14-s + (−0.172 + 0.324i)15-s + (−0.140 + 0.206i)16-s + (0.279 − 0.0942i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.02666 + 0.264900i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.02666 + 0.264900i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.856 - 0.515i)T \) |
| 3 | \( 1 + (-0.370 + 0.928i)T \) |
| 59 | \( 1 + (7.47 + 1.74i)T \) |
good | 5 | \( 1 + (1.41 + 0.153i)T + (4.88 + 1.07i)T^{2} \) |
| 7 | \( 1 + (-2.98 - 1.00i)T + (5.57 + 4.23i)T^{2} \) |
| 11 | \( 1 + (-2.18 - 3.23i)T + (-4.07 + 10.2i)T^{2} \) |
| 13 | \( 1 + (-3.44 + 3.26i)T + (0.703 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-1.15 + 0.388i)T + (13.5 - 10.2i)T^{2} \) |
| 19 | \( 1 + (-0.484 + 2.95i)T + (-18.0 - 6.06i)T^{2} \) |
| 23 | \( 1 + (0.198 - 0.715i)T + (-19.7 - 11.8i)T^{2} \) |
| 29 | \( 1 + (5.78 - 3.47i)T + (13.5 - 25.6i)T^{2} \) |
| 31 | \( 1 + (0.539 + 3.28i)T + (-29.3 + 9.89i)T^{2} \) |
| 37 | \( 1 + (-0.202 - 3.73i)T + (-36.7 + 4.00i)T^{2} \) |
| 41 | \( 1 + (1.29 + 4.65i)T + (-35.1 + 21.1i)T^{2} \) |
| 43 | \( 1 + (4.27 - 6.31i)T + (-15.9 - 39.9i)T^{2} \) |
| 47 | \( 1 + (8.23 - 0.895i)T + (45.9 - 10.1i)T^{2} \) |
| 53 | \( 1 + (2.61 - 1.99i)T + (14.1 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.52 - 2.11i)T + (28.5 + 53.8i)T^{2} \) |
| 67 | \( 1 + (-0.226 + 4.17i)T + (-66.6 - 7.24i)T^{2} \) |
| 71 | \( 1 + (-13.5 + 1.47i)T + (69.3 - 15.2i)T^{2} \) |
| 73 | \( 1 + (-3.86 - 4.54i)T + (-11.8 + 72.0i)T^{2} \) |
| 79 | \( 1 + (5.26 + 13.2i)T + (-57.3 + 54.3i)T^{2} \) |
| 83 | \( 1 + (-3.07 - 1.42i)T + (53.7 + 63.2i)T^{2} \) |
| 89 | \( 1 + (0.392 - 0.236i)T + (41.6 - 78.6i)T^{2} \) |
| 97 | \( 1 + (4.23 - 4.98i)T + (-15.6 - 95.7i)T^{2} \) |
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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
−11.63297757405005720625419884432, −11.04008699302717686198195031423, −9.463334572116202776980087256255, −8.290561126004005395180512459523, −7.79007492972866882265935537763, −6.78926846207104079335153239091, −5.59448273239318538481330295252, −4.56165981306967855374747201893, −3.37946449678972033200825863853, −1.72869848672394035347323587512,
1.61579712809850948848093730659, 3.59010450752569302078601873551, 4.06714354806145797717598957522, 5.30445943000924868338275707029, 6.44634714175897198707934025514, 7.82624630725474270168508986766, 8.592241395247414160250191085093, 9.696945080427124496677351502302, 10.97319025970856633365920212170, 11.29329001797178422163397509933