L(s) = 1 | + (0.256 + 8.99i)3-s + (−7.67 + 4.43i)5-s + (−30.9 + 53.5i)7-s + (−80.8 + 4.61i)9-s + (94.7 + 54.7i)11-s + (77.8 + 134. i)13-s + (−41.8 − 67.9i)15-s − 395. i·17-s + 140.·19-s + (−490. − 264. i)21-s + (802. − 463. i)23-s + (−273. + 473. i)25-s + (−62.3 − 726. i)27-s + (323. + 186. i)29-s + (521. + 903. i)31-s + ⋯ |
L(s) = 1 | + (0.0285 + 0.999i)3-s + (−0.307 + 0.177i)5-s + (−0.631 + 1.09i)7-s + (−0.998 + 0.0570i)9-s + (0.783 + 0.452i)11-s + (0.460 + 0.798i)13-s + (−0.186 − 0.302i)15-s − 1.37i·17-s + 0.388·19-s + (−1.11 − 0.599i)21-s + (1.51 − 0.876i)23-s + (−0.437 + 0.757i)25-s + (−0.0854 − 0.996i)27-s + (0.385 + 0.222i)29-s + (0.543 + 0.940i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.395 - 0.918i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.658627 + 1.00016i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.658627 + 1.00016i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.256 - 8.99i)T \) |
good | 5 | \( 1 + (7.67 - 4.43i)T + (312.5 - 541. i)T^{2} \) |
| 7 | \( 1 + (30.9 - 53.5i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-94.7 - 54.7i)T + (7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-77.8 - 134. i)T + (-1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 + 395. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 140.T + 1.30e5T^{2} \) |
| 23 | \( 1 + (-802. + 463. i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-323. - 186. i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-521. - 903. i)T + (-4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 194.T + 1.87e6T^{2} \) |
| 41 | \( 1 + (2.34e3 - 1.35e3i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (167. - 290. i)T + (-1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-2.46e3 - 1.42e3i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + 2.76e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (4.35e3 - 2.51e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-3.52e3 + 6.10e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (3.43e3 + 5.95e3i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 - 821. iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.09e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (3.78e3 - 6.55e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-6.77e3 - 3.91e3i)T + (2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + 1.28e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-1.89e3 + 3.27e3i)T + (-4.42e7 - 7.66e7i)T^{2} \) |
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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
−15.92189822661643308350699551019, −15.13305897070850996033696219115, −13.96131331296397124215022305446, −12.17154957449262183513134288067, −11.23151124490415729773004093541, −9.582478011906595605011515634291, −8.851058862585644979385790173690, −6.65624242790857659934643786170, −4.92039720834344289734560014032, −3.12279790009191266401488123388,
0.897499615307426482898906792143, 3.58584229886317101912470095279, 6.08755526591299637991715175892, 7.36296222142253873411879556290, 8.650287872900836555957864606195, 10.46949619308018215256541617667, 11.81412770996647847153293016213, 13.08456786798556711971891271953, 13.78921953381448328523581287282, 15.28433512690721603302267058275