L(s) = 1 | + 1.90·3-s + (2.41 − 4.37i)5-s − 3.95·7-s − 5.35·9-s + 6.18i·11-s + 4.94i·13-s + (4.60 − 8.35i)15-s + 22.4i·17-s − 10.3i·19-s − 7.54·21-s − 39.6·23-s + (−13.3 − 21.1i)25-s − 27.4·27-s + 30.4·29-s − 24.7i·31-s + ⋯ |
L(s) = 1 | + 0.636·3-s + (0.482 − 0.875i)5-s − 0.565·7-s − 0.595·9-s + 0.562i·11-s + 0.380i·13-s + (0.306 − 0.557i)15-s + 1.31i·17-s − 0.546i·19-s − 0.359·21-s − 1.72·23-s + (−0.534 − 0.845i)25-s − 1.01·27-s + 1.04·29-s − 0.797i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.875 - 0.482i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.875 - 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2441281013\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2441281013\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.41 + 4.37i)T \) |
good | 3 | \( 1 - 1.90T + 9T^{2} \) |
| 7 | \( 1 + 3.95T + 49T^{2} \) |
| 11 | \( 1 - 6.18iT - 121T^{2} \) |
| 13 | \( 1 - 4.94iT - 169T^{2} \) |
| 17 | \( 1 - 22.4iT - 289T^{2} \) |
| 19 | \( 1 + 10.3iT - 361T^{2} \) |
| 23 | \( 1 + 39.6T + 529T^{2} \) |
| 29 | \( 1 - 30.4T + 841T^{2} \) |
| 31 | \( 1 + 24.7iT - 961T^{2} \) |
| 37 | \( 1 + 24.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 31.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + 52.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 13.1T + 2.20e3T^{2} \) |
| 53 | \( 1 - 17.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 104. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 57.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 99.3T + 4.48e3T^{2} \) |
| 71 | \( 1 + 16.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 96.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 139. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 91.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 94.7T + 7.92e3T^{2} \) |
| 97 | \( 1 + 143. iT - 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.821475915352446868462991815670, −8.851932474448054635922866428056, −8.454796555185752362315721032940, −7.55842120027496582389152129673, −6.32489948947264692033684966501, −5.79708915512075762983788337577, −4.59542688403428179676447941118, −3.78848941191962502153086929669, −2.54438514782082840162566020747, −1.64514757462521645303786574165,
0.05769484764757730370375427082, 1.94616044738780020532752705724, 3.09189380055268303774228562458, 3.35416491789581462596596796368, 4.94728162058352559817189567739, 6.01084952360843644452435078007, 6.52751402013410115467609291089, 7.61570585520482273450546283907, 8.311892167748876869712106088385, 9.184623553093122798532525792142