| L(s) = 1 | + (−0.309 + 0.951i)2-s + (0.978 + 0.207i)3-s + (−0.809 − 0.587i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−2.21 + 0.987i)7-s + (0.809 − 0.587i)8-s + (0.913 + 0.406i)9-s + (0.978 − 0.207i)10-s + (0.123 − 1.17i)11-s + (−0.669 − 0.743i)12-s + (3.35 − 3.72i)13-s + (−0.253 − 2.41i)14-s + (−0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (−0.409 − 3.89i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (0.564 + 0.120i)3-s + (−0.404 − 0.293i)4-s + (−0.223 − 0.387i)5-s + (−0.204 + 0.353i)6-s + (−0.838 + 0.373i)7-s + (0.286 − 0.207i)8-s + (0.304 + 0.135i)9-s + (0.309 − 0.0657i)10-s + (0.0373 − 0.355i)11-s + (−0.193 − 0.214i)12-s + (0.929 − 1.03i)13-s + (−0.0678 − 0.645i)14-s + (−0.0797 − 0.245i)15-s + (0.0772 + 0.237i)16-s + (−0.0993 − 0.944i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12855 - 0.397427i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12855 - 0.397427i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 3 | \( 1 + (-0.978 - 0.207i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (-4.34 - 3.47i)T \) |
| good | 7 | \( 1 + (2.21 - 0.987i)T + (4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (-0.123 + 1.17i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (-3.35 + 3.72i)T + (-1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (0.409 + 3.89i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (4.85 + 5.39i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (1.97 - 1.43i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.450 + 1.38i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-4.71 + 8.16i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.38 + 0.931i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-3.69 - 4.10i)T + (-4.49 + 42.7i)T^{2} \) |
| 47 | \( 1 + (3.65 + 11.2i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (2.56 + 1.14i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (4.14 + 0.880i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 - 15.5T + 61T^{2} \) |
| 67 | \( 1 + (-5.37 - 9.31i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (11.9 + 5.31i)T + (47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (-1.34 + 12.7i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (0.640 + 6.09i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-4.23 + 0.900i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (-2.85 - 2.07i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (10.7 + 7.83i)T + (29.9 + 92.2i)T^{2} \) |
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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
−9.687240431640732382146108934606, −8.959312794156212245710051040912, −8.433822587650850119501519490949, −7.55552313343817628064062218244, −6.56360892851420807017152669592, −5.78692174139815678742784401876, −4.72946393763061871218625785073, −3.63835361630014644534914613356, −2.58502992126169173479069544919, −0.58941152778100798390045045465,
1.51253645214405529603646420662, 2.66856276750934619359156858788, 3.87240622058044656026189103814, 4.21198983736787624699547852561, 6.17232992824678469124129608477, 6.66975110719861531149991040145, 7.937772638526694578966948553288, 8.471982982268353946173180369980, 9.498573580790449884785074780271, 10.10941255066544915092400097418
