L(s) = 1 | − 2·2-s − 2.23i·5-s + (−2 + 6.70i)7-s + 8·8-s + 4.47i·10-s + 11-s − 20.1i·13-s + (4 − 13.4i)14-s − 16·16-s + 6.70i·17-s + 13.4i·19-s − 2·22-s − 8·23-s − 5.00·25-s + 40.2i·26-s + ⋯ |
L(s) = 1 | − 2-s − 0.447i·5-s + (−0.285 + 0.958i)7-s + 8-s + 0.447i·10-s + 0.0909·11-s − 1.54i·13-s + (0.285 − 0.958i)14-s − 16-s + 0.394i·17-s + 0.706i·19-s − 0.0909·22-s − 0.347·23-s − 0.200·25-s + 1.54i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.285i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0153607 + 0.105284i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0153607 + 0.105284i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 + (2 - 6.70i)T \) |
good | 2 | \( 1 + 2T + 4T^{2} \) |
| 11 | \( 1 - T + 121T^{2} \) |
| 13 | \( 1 + 20.1iT - 169T^{2} \) |
| 17 | \( 1 - 6.70iT - 289T^{2} \) |
| 19 | \( 1 - 13.4iT - 361T^{2} \) |
| 23 | \( 1 + 8T + 529T^{2} \) |
| 29 | \( 1 + 41T + 841T^{2} \) |
| 31 | \( 1 - 40.2iT - 961T^{2} \) |
| 37 | \( 1 + 28T + 1.36e3T^{2} \) |
| 41 | \( 1 + 13.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 82T + 1.84e3T^{2} \) |
| 47 | \( 1 + 20.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 74T + 2.80e3T^{2} \) |
| 59 | \( 1 - 93.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 80.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 14T + 5.04e3T^{2} \) |
| 73 | \( 1 + 67.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 19T + 6.24e3T^{2} \) |
| 83 | \( 1 + 93.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 107. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 60.3iT - 9.40e3T^{2} \) |
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
−11.84203424991169838271414540858, −10.54553527187164359757110465447, −9.953496362739849879256930202707, −8.895942094570851454762338479792, −8.372050483188912683321698182252, −7.43392777070978793591519495422, −5.92056777180052733090108045347, −5.02279514388090544641428924112, −3.41737939146948647881093911051, −1.65480455906079568924477226587,
0.07209804574946439744557187539, 1.80452642677742445078820855679, 3.72714961700696734284648483796, 4.76048845611138046000366308947, 6.55934953317910651612077696776, 7.23775219911217240731815787989, 8.195361890838903567322890191856, 9.456194152073651115681409118674, 9.727827629813074115294914447336, 10.97741909588005109914613529969