L(s) = 1 | + (0.00657 − 1.99i)2-s + (0.443 + 0.255i)3-s + (−3.99 − 0.0262i)4-s + (1.99 − 3.44i)5-s + (0.514 − 0.884i)6-s − 9.66i·7-s + (−0.0788 + 7.99i)8-s + (−4.36 − 7.56i)9-s + (−6.88 − 4.00i)10-s + 2.62i·11-s + (−1.76 − 1.03i)12-s + (11.8 + 20.4i)13-s + (−19.3 − 0.0635i)14-s + (1.76 − 1.01i)15-s + (15.9 + 0.210i)16-s + (−3.40 + 5.90i)17-s + ⋯ |
L(s) = 1 | + (0.00328 − 0.999i)2-s + (0.147 + 0.0852i)3-s + (−0.999 − 0.00657i)4-s + (0.398 − 0.689i)5-s + (0.0857 − 0.147i)6-s − 1.38i·7-s + (−0.00985 + 0.999i)8-s + (−0.485 − 0.840i)9-s + (−0.688 − 0.400i)10-s + 0.238i·11-s + (−0.147 − 0.0862i)12-s + (0.907 + 1.57i)13-s + (−1.38 − 0.00453i)14-s + (0.117 − 0.0678i)15-s + (0.999 + 0.0131i)16-s + (−0.200 + 0.347i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.461 + 0.887i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.461 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.632561 - 1.04226i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.632561 - 1.04226i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.00657 + 1.99i)T \) |
| 19 | \( 1 + (-12.0 + 14.7i)T \) |
good | 3 | \( 1 + (-0.443 - 0.255i)T + (4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (-1.99 + 3.44i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + 9.66iT - 49T^{2} \) |
| 11 | \( 1 - 2.62iT - 121T^{2} \) |
| 13 | \( 1 + (-11.8 - 20.4i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + (3.40 - 5.90i)T + (-144.5 - 250. i)T^{2} \) |
| 23 | \( 1 + (-17.2 + 9.96i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-0.445 - 0.772i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 - 59.6iT - 961T^{2} \) |
| 37 | \( 1 + 7.80T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-15.9 + 27.6i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (6.09 + 3.51i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (47.7 - 27.5i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (33.4 + 57.9i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-51.3 - 29.6i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (1.23 + 2.14i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-36.4 + 21.0i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (88.6 + 51.1i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (7.82 - 13.5i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-63.6 - 36.7i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 18.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (35.4 + 61.4i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (39.9 - 69.2i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−13.74121663147010181277380657885, −12.86969142615539758174218141084, −11.62731375968671870192364839266, −10.67375179119292734937037765232, −9.374747055652684206332659173962, −8.710728018748123892870553617489, −6.77265593503613289547176016622, −4.80902670577060442322202067766, −3.59217343933997716848891317022, −1.22775444372613298787100067357,
2.97247782639393009171419061880, 5.40154448720781128249122022933, 6.06983022324867110219765928032, 7.77330695094379031911358596110, 8.628206071992769341333766757090, 9.921605753574333865923034932284, 11.26678015050797071127434246455, 12.85409816839466541313816089865, 13.71572431913369374352355236802, 14.78833765376522954704929791257