L(s) = 1 | − 3·3-s + (−6.39 + 3.69i)5-s + (−3.39 + 5.88i)7-s + 9·9-s + (5.29 + 3.05i)11-s + (8.39 + 14.5i)13-s + (19.1 − 11.0i)15-s + 25.1i·17-s − 17.5·19-s + (10.1 − 17.6i)21-s + (−12.3 + 7.15i)23-s + (14.7 − 25.6i)25-s − 27·27-s + (−16.1 − 9.35i)29-s + (−23.3 − 40.5i)31-s + ⋯ |
L(s) = 1 | − 3-s + (−1.27 + 0.738i)5-s + (−0.485 + 0.841i)7-s + 9-s + (0.481 + 0.278i)11-s + (0.646 + 1.11i)13-s + (1.27 − 0.738i)15-s + 1.48i·17-s − 0.926·19-s + (0.485 − 0.841i)21-s + (−0.539 + 0.311i)23-s + (0.591 − 1.02i)25-s − 27-s + (−0.558 − 0.322i)29-s + (−0.754 − 1.30i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2321524320\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2321524320\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
good | 5 | \( 1 + (6.39 - 3.69i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (3.39 - 5.88i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-5.29 - 3.05i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-8.39 - 14.5i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 25.1iT - 289T^{2} \) |
| 19 | \( 1 + 17.5T + 361T^{2} \) |
| 23 | \( 1 + (12.3 - 7.15i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (16.1 + 9.35i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (23.3 + 40.5i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 49.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + (34.5 - 19.9i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-22.0 + 38.2i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (28.8 + 16.6i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 10.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (14.2 - 8.25i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-10.6 + 18.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-43.4 - 75.3i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 30.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 48.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-55.7 + 96.6i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (85.0 + 49.1i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 75.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-70.2 + 121. i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−11.37044836152985448410250414595, −10.42463789882155626792670482796, −9.408147805155981254307733046856, −8.362834155522881114019828123137, −7.34960226281422822109811851782, −6.41293590922224388091077974071, −5.91927850587679058184439227326, −4.23563988341693451252665424927, −3.78551219304738798781646494447, −1.94701430534579390970176225435,
0.13172050777226013689183905538, 0.971262284789838333069819944982, 3.44296817418871692739443809990, 4.27325879332092720008578462445, 5.17320620845654811577059928492, 6.34584305542696086593599559937, 7.25040704541379506212981675515, 8.030667746807309788959209014442, 9.076513948006652075988417868002, 10.16765645829244065422338583918