| L(s) = 1 | + (0.362 + 1.11i)2-s + (−0.978 + 0.207i)3-s + (0.503 − 0.365i)4-s + (−2.03 + 3.53i)5-s + (−0.586 − 1.01i)6-s + (1.97 + 0.881i)7-s + (2.49 + 1.80i)8-s + (0.913 − 0.406i)9-s + (−4.68 − 0.995i)10-s + (−0.660 − 6.28i)11-s + (−0.416 + 0.462i)12-s + (−1.09 − 1.21i)13-s + (−0.265 + 2.52i)14-s + (1.26 − 3.88i)15-s + (−0.731 + 2.25i)16-s + (0.198 − 1.89i)17-s + ⋯ |
| L(s) = 1 | + (0.256 + 0.789i)2-s + (−0.564 + 0.120i)3-s + (0.251 − 0.182i)4-s + (−0.912 + 1.58i)5-s + (−0.239 − 0.414i)6-s + (0.748 + 0.333i)7-s + (0.880 + 0.639i)8-s + (0.304 − 0.135i)9-s + (−1.48 − 0.314i)10-s + (−0.199 − 1.89i)11-s + (−0.120 + 0.133i)12-s + (−0.303 − 0.337i)13-s + (−0.0710 + 0.676i)14-s + (0.325 − 1.00i)15-s + (−0.182 + 0.562i)16-s + (0.0482 − 0.459i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0546 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0546 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.728608 + 0.689849i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.728608 + 0.689849i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.978 - 0.207i)T \) |
| 31 | \( 1 + (-0.846 - 5.50i)T \) |
| good | 2 | \( 1 + (-0.362 - 1.11i)T + (-1.61 + 1.17i)T^{2} \) |
| 5 | \( 1 + (2.03 - 3.53i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.97 - 0.881i)T + (4.68 + 5.20i)T^{2} \) |
| 11 | \( 1 + (0.660 + 6.28i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (1.09 + 1.21i)T + (-1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.198 + 1.89i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-0.411 + 0.456i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (-3.79 - 2.75i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.358 + 1.10i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-0.418 - 0.724i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.72 + 1.00i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (2.75 - 3.05i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (-2.53 + 7.79i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (8.67 - 3.86i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (-1.41 + 0.300i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + 7.03T + 61T^{2} \) |
| 67 | \( 1 + (-4.32 + 7.49i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.60 - 1.60i)T + (47.5 - 52.7i)T^{2} \) |
| 73 | \( 1 + (-0.503 - 4.79i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (0.0545 - 0.519i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (6.84 + 1.45i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + (-4.12 + 2.99i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.05 + 2.94i)T + (29.9 - 92.2i)T^{2} \) |
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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
−14.53199406399136976248581217993, −13.73059917022325396196168870252, −11.67528619386715377429738047996, −11.18862576955040502865810544791, −10.47395053742789582370800093609, −8.276071880726657829082280876341, −7.30132902732501752459436185030, −6.29002635881527670687313848373, −5.15553873835332414664433214644, −3.16093008651708888007459829391,
1.60987869793316050319257624188, 4.29815322130632804082984975633, 4.82614639029586008088436120351, 7.18598175708553085382392912878, 8.036933122073831594914949611039, 9.633428631568449389685121448146, 10.93949281895193073382936108757, 11.92253033526053784950487500380, 12.46094027444389934528223285869, 13.14088756196423891053029702150
