L(s) = 1 | + (−2.08 + 1.20i)2-s + (0.888 + 1.53i)3-s + (1.88 − 3.26i)4-s + 0.706i·5-s + (−3.69 − 2.13i)6-s + 4.25i·8-s + (−0.0791 + 0.137i)9-s + (−0.848 − 1.46i)10-s + (4.66 − 2.69i)11-s + 6.69·12-s + (−0.746 + 3.52i)13-s + (−1.08 + 0.627i)15-s + (−1.33 − 2.31i)16-s + (2.12 − 3.68i)17-s − 0.380i·18-s + (−2.40 − 1.38i)19-s + ⋯ |
L(s) = 1 | + (−1.47 + 0.849i)2-s + (0.513 + 0.888i)3-s + (0.942 − 1.63i)4-s + 0.316i·5-s + (−1.50 − 0.871i)6-s + 1.50i·8-s + (−0.0263 + 0.0456i)9-s + (−0.268 − 0.464i)10-s + (1.40 − 0.812i)11-s + 1.93·12-s + (−0.207 + 0.978i)13-s + (−0.280 + 0.162i)15-s + (−0.333 − 0.578i)16-s + (0.515 − 0.892i)17-s − 0.0895i·18-s + (−0.551 − 0.318i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.564483 + 0.767162i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.564483 + 0.767162i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (0.746 - 3.52i)T \) |
good | 2 | \( 1 + (2.08 - 1.20i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.888 - 1.53i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 - 0.706iT - 5T^{2} \) |
| 11 | \( 1 + (-4.66 + 2.69i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.12 + 3.68i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.40 + 1.38i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.35 - 5.81i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.27 + 2.21i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 8.74iT - 31T^{2} \) |
| 37 | \( 1 + (1.18 - 0.682i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.91 + 2.25i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.78 + 6.54i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 0.0870iT - 47T^{2} \) |
| 53 | \( 1 - 7.04T + 53T^{2} \) |
| 59 | \( 1 + (5.41 + 3.12i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.40 - 11.0i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (9.09 - 5.25i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (9.03 + 5.21i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 10.1iT - 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 - 3.20iT - 83T^{2} \) |
| 89 | \( 1 + (1.75 - 1.01i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.62 - 2.67i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.46049813374540912166531784363, −9.599064715486696600491222926251, −8.919541557868467702311830542705, −8.772813699039879029496098135850, −7.21079559315608976332784770041, −6.84626633368087652248092690746, −5.71141904273754023697597489188, −4.29435250442663339830815351009, −3.14089306609012690805132278459, −1.24013701841236827611023724002,
1.01449087543347307547984978665, 1.91760005866861310135576992368, 3.00403115669968970027781138503, 4.44013047932770655764376037421, 6.23104727523305935577453699656, 7.27109136993487642828095951006, 7.912834274651699003109390059536, 8.675340709006182613403713226780, 9.330455572115103340047930847090, 10.28904750858393271276262850158