L(s) = 1 | − 4.00·2-s + (34.4 + 31.5i)3-s − 111.·4-s − 109. i·5-s + (−138. − 126. i)6-s − 1.22e3i·7-s + 960.·8-s + (192. + 2.17e3i)9-s + 437. i·10-s + (4.40e3 − 355. i)11-s + (−3.86e3 − 3.53e3i)12-s − 2.12e3i·13-s + 4.89e3i·14-s + (3.45e3 − 3.77e3i)15-s + 1.04e4·16-s + 2.38e4·17-s + ⋯ |
L(s) = 1 | − 0.353·2-s + (0.737 + 0.675i)3-s − 0.874·4-s − 0.391i·5-s + (−0.260 − 0.238i)6-s − 1.34i·7-s + 0.663·8-s + (0.0879 + 0.996i)9-s + 0.138i·10-s + (0.996 − 0.0804i)11-s + (−0.645 − 0.590i)12-s − 0.267i·13-s + 0.476i·14-s + (0.264 − 0.288i)15-s + 0.640·16-s + 1.17·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(1.43148 - 0.490977i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43148 - 0.490977i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-34.4 - 31.5i)T \) |
| 11 | \( 1 + (-4.40e3 + 355. i)T \) |
good | 2 | \( 1 + 4.00T + 128T^{2} \) |
| 5 | \( 1 + 109. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + 1.22e3iT - 8.23e5T^{2} \) |
| 13 | \( 1 + 2.12e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 2.38e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.67e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 4.79e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 1.69e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 7.43e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.77e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.45e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.64e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 1.20e6iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 1.78e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 3.88e5iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 2.73e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 4.55e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.64e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 + 2.47e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 4.90e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 - 2.93e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.83e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 - 6.00e6T + 8.07e13T^{2} \) |
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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
−14.81615887326491790733128578379, −13.92626817953228175592857667293, −12.98589370189710092257564615794, −10.84626722194754903648936295676, −9.718891574401261048948635769273, −8.770964139363841672297696263369, −7.44965151171327053393668812570, −4.80699195505542376774063798365, −3.69375783852385943275009860212, −0.861486670923231932241685557539,
1.55966900467007246557130897899, 3.53643139236822007008908452668, 5.84890302313285472117601854057, 7.66752281194962653067823761837, 8.865487117925465768642557108674, 9.707049040802470001812485978151, 11.87536592688004216333653990773, 12.85044867871167964250859214233, 14.34209251767031239066866141758, 14.79030378433692053632131933246