L(s) = 1 | − 6.70i·2-s − 29.0·4-s + 41.0·5-s − 6.70·7-s + 87.3i·8-s − 275. i·10-s + 21.2i·11-s − 73.2i·13-s + 45.0i·14-s + 121.·16-s − 76.4·17-s + 439.·19-s − 1.19e3·20-s + 142.·22-s + 164. i·23-s + ⋯ |
L(s) = 1 | − 1.67i·2-s − 1.81·4-s + 1.64·5-s − 0.136·7-s + 1.36i·8-s − 2.75i·10-s + 0.175i·11-s − 0.433i·13-s + 0.229i·14-s + 0.476·16-s − 0.264·17-s + 1.21·19-s − 2.98·20-s + 0.295·22-s + 0.311i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0296i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.434123061\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.434123061\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 59 | \( 1 + (-3.47e3 + 103. i)T \) |
good | 2 | \( 1 + 6.70iT - 16T^{2} \) |
| 5 | \( 1 - 41.0T + 625T^{2} \) |
| 7 | \( 1 + 6.70T + 2.40e3T^{2} \) |
| 11 | \( 1 - 21.2iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 73.2iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 76.4T + 8.35e4T^{2} \) |
| 19 | \( 1 - 439.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 164. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 788.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 754. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.70e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 72.3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 2.69e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.90e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.17e3T + 7.89e6T^{2} \) |
| 61 | \( 1 + 2.04e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 796. iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 4.86e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 1.82e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 7.08e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 4.42e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 8.04e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 2.35e3iT - 8.85e7T^{2} \) |
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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
−9.853942100419603917318739626706, −9.491884034680119902149981188287, −8.523766444046212731309336458606, −6.99491311739934605399434497026, −5.75747417821251430712865911704, −4.96830704281953342815781059611, −3.59730555125863468990028474203, −2.54944385633942153530406151037, −1.77822916643467119732285665222, −0.65861238985265980643337822198,
1.28343044164461353504178487074, 2.84256827836518700681732359918, 4.64851236561140060220313880430, 5.37301323838136782034514698884, 6.30109803838491911698359338683, 6.70772057437179264670075207660, 7.87988482909949158716975395490, 8.839291064600661796665165415211, 9.524058125788799011806845256303, 10.20498637851083067706970854147