L(s) = 1 | + (−2.90 − 0.751i)3-s + (−1.38 − 4.80i)5-s + 3.03i·7-s + (7.86 + 4.36i)9-s + 11.4·11-s − 16.9·13-s + (0.413 + 14.9i)15-s − 18.1·17-s − 16.9i·19-s + (2.28 − 8.82i)21-s + 22.3·23-s + (−21.1 + 13.3i)25-s + (−19.5 − 18.5i)27-s − 17.8·29-s − 16.9·31-s + ⋯ |
L(s) = 1 | + (−0.968 − 0.250i)3-s + (−0.277 − 0.960i)5-s + 0.434i·7-s + (0.874 + 0.485i)9-s + 1.04·11-s − 1.30·13-s + (0.0275 + 0.999i)15-s − 1.06·17-s − 0.891i·19-s + (0.108 − 0.420i)21-s + 0.971·23-s + (−0.846 + 0.532i)25-s + (−0.724 − 0.688i)27-s − 0.617·29-s − 0.547·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.232 - 0.972i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5606835611\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5606835611\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.90 + 0.751i)T \) |
| 5 | \( 1 + (1.38 + 4.80i)T \) |
good | 7 | \( 1 - 3.03iT - 49T^{2} \) |
| 11 | \( 1 - 11.4T + 121T^{2} \) |
| 13 | \( 1 + 16.9T + 169T^{2} \) |
| 17 | \( 1 + 18.1T + 289T^{2} \) |
| 19 | \( 1 + 16.9iT - 361T^{2} \) |
| 23 | \( 1 - 22.3T + 529T^{2} \) |
| 29 | \( 1 + 17.8T + 841T^{2} \) |
| 31 | \( 1 + 16.9T + 961T^{2} \) |
| 37 | \( 1 + 1.21T + 1.36e3T^{2} \) |
| 41 | \( 1 - 42.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.98T + 1.84e3T^{2} \) |
| 47 | \( 1 + 52.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 8.58iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 93.0T + 3.48e3T^{2} \) |
| 61 | \( 1 - 80.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 38.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 109. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 28.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 56.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 126. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 126. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 153. iT - 9.40e3T^{2} \) |
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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
−9.856977003690791458321487236443, −9.237022991505513638551260428536, −8.436426375578373330099215602826, −7.22978705710057479439743160201, −6.71149153486538658873625348310, −5.51045237742711155088232818183, −4.87008094719768437225693858515, −4.07500484790690435027620598608, −2.30396452751861033160193865074, −1.01982229447339190965991695141,
0.24233668182105026971206751389, 1.94362085773703843750377291674, 3.49178854839801549351388998355, 4.28059912489442567351049183041, 5.28531920308933987729860089503, 6.41644702947363085889089897042, 6.95305441191029061379241588237, 7.63576509109897976973338411275, 9.071293477199109311715050802558, 9.831780516806109730782151480160