L(s) = 1 | − 4.94i·2-s + (182. + 160. i)3-s + 9.99e2·4-s + 1.39e3i·5-s + (790. − 903. i)6-s − 5.51e3·7-s − 9.99e3i·8-s + (7.79e3 + 5.85e4i)9-s + 6.90e3·10-s + 7.67e4i·11-s + (1.82e5 + 1.60e5i)12-s + 4.49e5·13-s + 2.72e4i·14-s + (−2.23e5 + 2.55e5i)15-s + 9.74e5·16-s + 1.42e6i·17-s + ⋯ |
L(s) = 1 | − 0.154i·2-s + (0.752 + 0.658i)3-s + 0.976·4-s + 0.447i·5-s + (0.101 − 0.116i)6-s − 0.328·7-s − 0.305i·8-s + (0.131 + 0.991i)9-s + 0.0690·10-s + 0.476i·11-s + (0.734 + 0.643i)12-s + 1.21·13-s + 0.0506i·14-s + (−0.294 + 0.336i)15-s + 0.929·16-s + 1.00i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(2.38328 + 1.08088i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.38328 + 1.08088i\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-182. - 160. i)T \) |
| 5 | \( 1 - 1.39e3iT \) |
good | 2 | \( 1 + 4.94iT - 1.02e3T^{2} \) |
| 7 | \( 1 + 5.51e3T + 2.82e8T^{2} \) |
| 11 | \( 1 - 7.67e4iT - 2.59e10T^{2} \) |
| 13 | \( 1 - 4.49e5T + 1.37e11T^{2} \) |
| 17 | \( 1 - 1.42e6iT - 2.01e12T^{2} \) |
| 19 | \( 1 + 1.48e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + 8.24e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 + 3.07e7iT - 4.20e14T^{2} \) |
| 31 | \( 1 + 1.77e7T + 8.19e14T^{2} \) |
| 37 | \( 1 - 7.18e7T + 4.80e15T^{2} \) |
| 41 | \( 1 - 2.33e7iT - 1.34e16T^{2} \) |
| 43 | \( 1 + 1.37e7T + 2.16e16T^{2} \) |
| 47 | \( 1 + 4.03e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 + 4.99e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + 7.43e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 1.54e9T + 7.13e17T^{2} \) |
| 67 | \( 1 - 1.83e9T + 1.82e18T^{2} \) |
| 71 | \( 1 - 3.45e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 8.31e8T + 4.29e18T^{2} \) |
| 79 | \( 1 - 4.69e8T + 9.46e18T^{2} \) |
| 83 | \( 1 + 1.59e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 - 6.09e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + 2.89e9T + 7.37e19T^{2} \) |
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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
−16.72599126747585877866573848104, −15.57426540133522267464019572660, −14.69094190299633982591752835600, −12.98593956118860172747645496017, −11.10262821289527022655411259395, −10.05936043661154204579868611404, −8.200871257788894210727330673823, −6.40166173092985291935447085512, −3.76236182823639145991726241936, −2.18533207826646460269541274890,
1.35800903504548086155324583343, 3.18060726202574314852479056961, 6.14639106941958506529895742316, 7.60367655109067309015695647376, 9.074194445082913289981948390707, 11.19536439131664579232937941980, 12.65740274927595316353739006501, 13.92659019264758989478089030699, 15.45180632280677812027511838713, 16.48494359168388404266677155594