L(s) = 1 | − 2i·2-s − 2·4-s + 2i·7-s − 11-s + 4i·13-s + 4·14-s − 4·16-s − 2i·17-s + 2i·22-s + i·23-s + 8·26-s − 4i·28-s + 7·31-s + 8i·32-s − 4·34-s + ⋯ |
L(s) = 1 | − 1.41i·2-s − 4-s + 0.755i·7-s − 0.301·11-s + 1.10i·13-s + 1.06·14-s − 16-s − 0.485i·17-s + 0.426i·22-s + 0.208i·23-s + 1.56·26-s − 0.755i·28-s + 1.25·31-s + 1.41i·32-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.638542166\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.638542166\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 2iT - 2T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 + 3iT - 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 5T + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 - 7iT - 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 - 7iT - 97T^{2} \) |
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
−9.165084479317677098385494198288, −8.328148512145031190969761258208, −7.26801250680856411924011612348, −6.46507675682819691012914834694, −5.48839480308999129741362162432, −4.53131206918408403275588185363, −3.80720158844220703298532645371, −2.67544941524919290821879443709, −2.20193489479546506633400668487, −0.955373789394485655072388047414,
0.69904508467170606839526776832, 2.39339510032273414591697618803, 3.58582445546653028719375067578, 4.59718540584982485388701259771, 5.29958517375224659287044402422, 6.12413043052292813165365685578, 6.74198575776120250887442349566, 7.58060360641750635872273022794, 8.067347954202489738830910533190, 8.660696378935712273189641518391