L(s) = 1 | + (−1.92 + 0.625i)2-s + (1.54 − 2.12i)3-s + (1.69 − 1.23i)4-s + (2.01 + 0.965i)5-s + (−1.63 + 5.04i)6-s + (0.567 + 0.781i)7-s + (−0.113 + 0.156i)8-s + (−1.19 − 3.67i)9-s + (−4.48 − 0.596i)10-s − 5.49i·12-s + (4.30 − 1.39i)13-s + (−1.58 − 1.14i)14-s + (5.15 − 2.78i)15-s + (−1.17 + 3.61i)16-s + (3.17 + 1.03i)17-s + (4.60 + 6.33i)18-s + ⋯ |
L(s) = 1 | + (−1.36 + 0.442i)2-s + (0.889 − 1.22i)3-s + (0.847 − 0.615i)4-s + (0.901 + 0.431i)5-s + (−0.669 + 2.05i)6-s + (0.214 + 0.295i)7-s + (−0.0400 + 0.0551i)8-s + (−0.398 − 1.22i)9-s + (−1.41 − 0.188i)10-s − 1.58i·12-s + (1.19 − 0.387i)13-s + (−0.422 − 0.307i)14-s + (1.33 − 0.720i)15-s + (−0.293 + 0.903i)16-s + (0.769 + 0.250i)17-s + (1.08 + 1.49i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.22743 - 0.248116i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.22743 - 0.248116i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-2.01 - 0.965i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (1.92 - 0.625i)T + (1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (-1.54 + 2.12i)T + (-0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + (-0.567 - 0.781i)T + (-2.16 + 6.65i)T^{2} \) |
| 13 | \( 1 + (-4.30 + 1.39i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-3.17 - 1.03i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (2.65 + 1.92i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 3.36iT - 23T^{2} \) |
| 29 | \( 1 + (3.97 - 2.88i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.129 + 0.397i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (3.72 + 5.12i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-4.68 - 3.40i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 2.26iT - 43T^{2} \) |
| 47 | \( 1 + (-2.54 + 3.49i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (2.53 - 0.822i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (8.19 - 5.95i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.763 + 2.34i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + 9.60iT - 67T^{2} \) |
| 71 | \( 1 + (1.68 - 5.18i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.843 + 1.16i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.310 - 0.954i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-7.03 - 2.28i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 + (-2.87 + 0.932i)T + (78.4 - 57.0i)T^{2} \) |
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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
−10.40003394510335982785094392263, −9.290909474453089296252834197531, −8.810171650802304771151740869304, −7.997434148411938980709383515168, −7.30830863369612383481678939881, −6.49415219973163737094341155319, −5.66791634078736027301882405267, −3.46032407223529196009608440554, −2.11365852684418720315803219570, −1.25633552932798708442058878479,
1.38164916419370291053260005226, 2.61104100011539732953519748855, 3.91513277748520126163273366214, 4.96287861456768112553164687354, 6.23951359363013612672599565677, 7.78121120971204245385500547595, 8.522449942927563556549135123141, 9.134696209235206220302618734309, 9.671052351187947515293753639160, 10.49403318941712478876491747665