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Error: no document with id 267897781 found in table mf_hecke_traces.
| L(s) = 1 | − 128i·2-s − 1.63e4·4-s + (3.12e4 − 1.71e5i)5-s + 5.11e5i·7-s + 2.09e6i·8-s + (−2.20e7 − 4.00e6i)10-s + 1.99e7·11-s + 1.80e8i·13-s + 6.55e7·14-s + 2.68e8·16-s − 5.65e8i·17-s + 2.16e9·19-s + (−5.12e8 + 2.81e9i)20-s − 2.55e9i·22-s − 3.44e9i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (0.178 − 0.983i)5-s + 0.234i·7-s + 0.353i·8-s + (−0.695 − 0.126i)10-s + 0.308·11-s + 0.796i·13-s + 0.166·14-s + 0.250·16-s − 0.334i·17-s + 0.556·19-s + (−0.0894 + 0.491i)20-s − 0.218i·22-s − 0.210i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(1.253170766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.253170766\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 128iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-3.12e4 + 1.71e5i)T \) |
| good | 7 | \( 1 - 5.11e5iT - 4.74e12T^{2} \) |
| 11 | \( 1 - 1.99e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 1.80e8iT - 5.11e16T^{2} \) |
| 17 | \( 1 + 5.65e8iT - 2.86e18T^{2} \) |
| 19 | \( 1 - 2.16e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + 3.44e9iT - 2.66e20T^{2} \) |
| 29 | \( 1 - 5.88e10T + 8.62e21T^{2} \) |
| 31 | \( 1 + 1.31e11T + 2.34e22T^{2} \) |
| 37 | \( 1 + 1.01e12iT - 3.33e23T^{2} \) |
| 41 | \( 1 - 6.78e11T + 1.55e24T^{2} \) |
| 43 | \( 1 + 1.86e12iT - 3.17e24T^{2} \) |
| 47 | \( 1 - 2.65e12iT - 1.20e25T^{2} \) |
| 53 | \( 1 - 7.73e12iT - 7.31e25T^{2} \) |
| 59 | \( 1 - 1.23e13T + 3.65e26T^{2} \) |
| 61 | \( 1 + 2.87e13T + 6.02e26T^{2} \) |
| 67 | \( 1 + 6.21e13iT - 2.46e27T^{2} \) |
| 71 | \( 1 + 1.70e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + 9.44e13iT - 8.90e27T^{2} \) |
| 79 | \( 1 + 5.16e12T + 2.91e28T^{2} \) |
| 83 | \( 1 + 3.88e14iT - 6.11e28T^{2} \) |
| 89 | \( 1 - 6.49e13T + 1.74e29T^{2} \) |
| 97 | \( 1 + 4.24e14iT - 6.33e29T^{2} \) |
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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
−10.68640804403654752205900590169, −9.345185401240123335301947240616, −8.919213868956667916774896548657, −7.50664589665096535854990731040, −5.91640095209456389399571308004, −4.81975380625624555861558691522, −3.82734916982949854338895145860, −2.34396240641689163659343828462, −1.33331935536696541436105931322, −0.27321704020100043393682235038,
1.18612591118500163204755486661, 2.79322623524827563558036361298, 3.86496832525289434845524819863, 5.32356731400496886725338702197, 6.37074946115319658394210018077, 7.28237126162909605567023496021, 8.279344760101751313318595488296, 9.659772168325626424438701266347, 10.50788197770671459137002184597, 11.66328667157654859072907956502
