| L(s) = 1 | + (−0.220 − 1.53i)2-s + (0.909 + 0.415i)3-s + (−0.382 + 0.112i)4-s + (0.186 − 0.215i)5-s + (0.436 − 1.48i)6-s + (0.0250 − 2.64i)7-s + (−1.03 − 2.25i)8-s + (0.654 + 0.755i)9-s + (−0.370 − 0.238i)10-s + (−0.948 − 0.136i)11-s + (−0.394 − 0.0567i)12-s + (0.623 − 0.970i)13-s + (−4.06 + 0.544i)14-s + (0.258 − 0.118i)15-s + (−3.90 + 2.50i)16-s + (2.01 + 0.590i)17-s + ⋯ |
| L(s) = 1 | + (−0.155 − 1.08i)2-s + (0.525 + 0.239i)3-s + (−0.191 + 0.0561i)4-s + (0.0833 − 0.0961i)5-s + (0.178 − 0.606i)6-s + (0.00947 − 0.999i)7-s + (−0.364 − 0.797i)8-s + (0.218 + 0.251i)9-s + (−0.117 − 0.0753i)10-s + (−0.286 − 0.0411i)11-s + (−0.113 − 0.0163i)12-s + (0.172 − 0.269i)13-s + (−1.08 + 0.145i)14-s + (0.0668 − 0.0305i)15-s + (−0.975 + 0.626i)16-s + (0.487 + 0.143i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.551 + 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.551 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.767276 - 1.42802i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.767276 - 1.42802i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.909 - 0.415i)T \) |
| 7 | \( 1 + (-0.0250 + 2.64i)T \) |
| 23 | \( 1 + (0.598 + 4.75i)T \) |
| good | 2 | \( 1 + (0.220 + 1.53i)T + (-1.91 + 0.563i)T^{2} \) |
| 5 | \( 1 + (-0.186 + 0.215i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (0.948 + 0.136i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.623 + 0.970i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-2.01 - 0.590i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-1.02 + 0.299i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (6.80 + 1.99i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-9.11 + 4.16i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-2.02 + 1.75i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (4.47 + 3.88i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-7.02 - 3.20i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 10.3iT - 47T^{2} \) |
| 53 | \( 1 + (-1.25 - 1.95i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (1.73 - 2.70i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-4.20 - 9.20i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.584 + 0.0840i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (1.37 + 9.59i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-1.30 - 4.45i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (1.87 - 2.91i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-10.1 - 11.7i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (5.28 - 11.5i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-6.11 + 7.05i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.67271836369373202103916363619, −9.942696426175098582955104729118, −9.264557512776545044150828148993, −8.057686105483526029348311904509, −7.20540157184494728786884968306, −5.98224326609994047392532572051, −4.45311956729472348578767266598, −3.54879449254404681397431251079, −2.49733140143939444359811305024, −1.03112122174435966400725246961,
2.09632003161874955829325911667, 3.23599387201745155484934912893, 4.99613725236886579330043921150, 5.89659705533158010279341629234, 6.74656193085115411026968888831, 7.70429464241028712604860310084, 8.408114139514353864175427677557, 9.152277389734037669511301356607, 10.12994173291730418589117576559, 11.51462527886266522356940350962
