| L(s) = 1 | + (−0.859 − 1.12i)2-s + (−0.160 + 1.72i)3-s + (−0.524 + 1.93i)4-s + (2.32 − 1.08i)5-s + (2.07 − 1.30i)6-s + (0.387 − 2.19i)7-s + (2.61 − 1.06i)8-s + (−2.94 − 0.554i)9-s + (−3.21 − 1.67i)10-s + (3.07 + 1.43i)11-s + (−3.24 − 1.21i)12-s + (0.323 − 3.69i)13-s + (−2.80 + 1.45i)14-s + (1.49 + 4.18i)15-s + (−3.45 − 2.02i)16-s + (−4.87 + 2.81i)17-s + ⋯ |
| L(s) = 1 | + (−0.607 − 0.794i)2-s + (−0.0928 + 0.995i)3-s + (−0.262 + 0.965i)4-s + (1.03 − 0.484i)5-s + (0.847 − 0.531i)6-s + (0.146 − 0.830i)7-s + (0.925 − 0.378i)8-s + (−0.982 − 0.184i)9-s + (−1.01 − 0.531i)10-s + (0.925 + 0.431i)11-s + (−0.936 − 0.350i)12-s + (0.0896 − 1.02i)13-s + (−0.748 + 0.388i)14-s + (0.386 + 1.07i)15-s + (−0.862 − 0.505i)16-s + (−1.18 + 0.682i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.14939 - 0.331690i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14939 - 0.331690i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.859 + 1.12i)T \) |
| 3 | \( 1 + (0.160 - 1.72i)T \) |
| good | 5 | \( 1 + (-2.32 + 1.08i)T + (3.21 - 3.83i)T^{2} \) |
| 7 | \( 1 + (-0.387 + 2.19i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-3.07 - 1.43i)T + (7.07 + 8.42i)T^{2} \) |
| 13 | \( 1 + (-0.323 + 3.69i)T + (-12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (4.87 - 2.81i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.54 + 1.21i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-5.25 + 0.927i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.224 - 2.56i)T + (-28.5 + 5.03i)T^{2} \) |
| 31 | \( 1 + (-8.28 + 1.46i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.66 + 6.22i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-4.49 + 3.77i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (4.33 - 9.30i)T + (-27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (0.437 - 2.48i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (3.02 - 3.02i)T - 53iT^{2} \) |
| 59 | \( 1 + (-3.37 - 7.24i)T + (-37.9 + 45.1i)T^{2} \) |
| 61 | \( 1 + (7.89 + 11.2i)T + (-20.8 + 57.3i)T^{2} \) |
| 67 | \( 1 + (6.97 + 0.610i)T + (65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (7.24 - 4.18i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (7.40 + 4.27i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.05 - 2.44i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-4.52 + 0.396i)T + (81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (4.33 - 7.50i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.0 + 4.76i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.79466377128499886563772249315, −10.21191856058042714666823999751, −9.336950486591169621704753360181, −8.899841475831364166696149703647, −7.65658845699827532123503478418, −6.31188390641575607654167119513, −4.95071386214631871585827455916, −4.12144195997576761638545306426, −2.84684693651515555359243337440, −1.16929232555930686660148390111,
1.42370121175183467131278418703, 2.58420135311421556408890679525, 4.92423250885146187999653800737, 6.05431847019232940179962185865, 6.50614525823104900616695114452, 7.31919623099725429904722227432, 8.685674801153083628947733171175, 9.078589367751794595857753632172, 10.07462467525212588115098722322, 11.42815773602687784395421795297
