L(s) = 1 | − 1.77i·2-s + (−1.65 − 0.524i)3-s − 1.14·4-s − 3.73·5-s + (−0.929 + 2.92i)6-s + (−2.02 − 1.69i)7-s − 1.52i·8-s + (2.45 + 1.73i)9-s + 6.61i·10-s + 1.46i·11-s + (1.88 + 0.598i)12-s + 0.721i·13-s + (−3.01 + 3.59i)14-s + (6.15 + 1.95i)15-s − 4.97·16-s + 4.35·17-s + ⋯ |
L(s) = 1 | − 1.25i·2-s + (−0.953 − 0.302i)3-s − 0.570·4-s − 1.66·5-s + (−0.379 + 1.19i)6-s + (−0.766 − 0.642i)7-s − 0.537i·8-s + (0.816 + 0.577i)9-s + 2.09i·10-s + 0.441i·11-s + (0.544 + 0.172i)12-s + 0.200i·13-s + (−0.805 + 0.960i)14-s + (1.58 + 0.505i)15-s − 1.24·16-s + 1.05·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.134829 + 0.0391670i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.134829 + 0.0391670i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.65 + 0.524i)T \) |
| 7 | \( 1 + (2.02 + 1.69i)T \) |
| 23 | \( 1 - iT \) |
good | 2 | \( 1 + 1.77iT - 2T^{2} \) |
| 5 | \( 1 + 3.73T + 5T^{2} \) |
| 11 | \( 1 - 1.46iT - 11T^{2} \) |
| 13 | \( 1 - 0.721iT - 13T^{2} \) |
| 17 | \( 1 - 4.35T + 17T^{2} \) |
| 19 | \( 1 - 0.368iT - 19T^{2} \) |
| 29 | \( 1 - 2.64iT - 29T^{2} \) |
| 31 | \( 1 - 9.03iT - 31T^{2} \) |
| 37 | \( 1 + 4.69T + 37T^{2} \) |
| 41 | \( 1 - 1.09T + 41T^{2} \) |
| 43 | \( 1 + 9.97T + 43T^{2} \) |
| 47 | \( 1 + 4.98T + 47T^{2} \) |
| 53 | \( 1 + 10.0iT - 53T^{2} \) |
| 59 | \( 1 - 7.05T + 59T^{2} \) |
| 61 | \( 1 + 7.71iT - 61T^{2} \) |
| 67 | \( 1 + 3.88T + 67T^{2} \) |
| 71 | \( 1 - 12.2iT - 71T^{2} \) |
| 73 | \( 1 - 14.1iT - 73T^{2} \) |
| 79 | \( 1 + 9.18T + 79T^{2} \) |
| 83 | \( 1 + 1.74T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 - 10.2iT - 97T^{2} \) |
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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
−11.31151834179397980307989074185, −10.37381674187744286819426094422, −9.867500402177676657546384844650, −8.343965257126205015912688029506, −7.14788386013608859971187831413, −6.81354233112212410262025268166, −5.04403201478991767511544763628, −3.95996965780849469338653553969, −3.26022200817456265431163605136, −1.27078386575783689595178862828,
0.10712495819332755121885465232, 3.27294811432189559118350129038, 4.41226415169627045517520599343, 5.50537812235591376775605095689, 6.24011659927111225365102686957, 7.18890458218126932668334415940, 7.921792418763013078462505424467, 8.809582631752157194931963255084, 9.982554390584541018670471593536, 11.12813034561265567925215155544