L(s) = 1 | + (0.611 + 1.05i)2-s + (−0.866 + 0.5i)3-s + (0.251 − 0.436i)4-s + (2.22 − 0.247i)5-s + (−1.05 − 0.611i)6-s + (−0.997 + 1.72i)7-s + 3.06·8-s + (0.499 − 0.866i)9-s + (1.62 + 2.20i)10-s + (0.0539 − 0.0311i)11-s + 0.503i·12-s + (1.38 + 3.33i)13-s − 2.44·14-s + (−1.80 + 1.32i)15-s + (1.36 + 2.37i)16-s + (−5.17 − 2.99i)17-s + ⋯ |
L(s) = 1 | + (0.432 + 0.749i)2-s + (−0.499 + 0.288i)3-s + (0.125 − 0.218i)4-s + (0.993 − 0.110i)5-s + (−0.432 − 0.249i)6-s + (−0.377 + 0.653i)7-s + 1.08·8-s + (0.166 − 0.288i)9-s + (0.512 + 0.696i)10-s + (0.0162 − 0.00938i)11-s + 0.145i·12-s + (0.383 + 0.923i)13-s − 0.652·14-s + (−0.464 + 0.342i)15-s + (0.342 + 0.592i)16-s + (−1.25 − 0.725i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.33416 + 0.742051i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.33416 + 0.742051i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (-2.22 + 0.247i)T \) |
| 13 | \( 1 + (-1.38 - 3.33i)T \) |
good | 2 | \( 1 + (-0.611 - 1.05i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (0.997 - 1.72i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.0539 + 0.0311i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (5.17 + 2.99i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.14 + 0.661i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.713 - 0.411i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.04 + 7.00i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4.04iT - 31T^{2} \) |
| 37 | \( 1 + (2.72 + 4.71i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-9.94 + 5.73i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (7.52 + 4.34i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 9.45T + 47T^{2} \) |
| 53 | \( 1 + 1.39iT - 53T^{2} \) |
| 59 | \( 1 + (3.05 + 1.76i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.83 - 6.64i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.17 - 8.95i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.394 + 0.227i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 2.99T + 73T^{2} \) |
| 79 | \( 1 + 8.01T + 79T^{2} \) |
| 83 | \( 1 - 4.14T + 83T^{2} \) |
| 89 | \( 1 + (9.96 - 5.75i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.91 + 10.2i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.94745040260354179036927895134, −11.62368512033205792879242988343, −10.72780857732672071388035082374, −9.640763187686016667355074809768, −8.853586931240902128981549966590, −7.02902467726941389080499884201, −6.25833623887296654920635865485, −5.48721885815783514553557092211, −4.39556307775622640522000948324, −2.10578837123490216429596573013,
1.72667854402034709731635103851, 3.23151663808899212304212871310, 4.65093966063384990820540005753, 6.06410838905668899468729469847, 6.98982850740632337301156692327, 8.272504295431471294243440386406, 9.802762476115189051244106957596, 10.70787088973446640088953643964, 11.20726254511632286908590593336, 12.66930311744992675424042246168