L(s) = 1 | + (−0.5 + 0.866i)2-s + (1.65 − 0.517i)3-s + (−0.499 − 0.866i)4-s + 0.188i·5-s + (−0.377 + 1.69i)6-s + (−2.53 + 0.772i)7-s + 0.999·8-s + (2.46 − 1.71i)9-s + (−0.163 − 0.0944i)10-s + (2.99 − 5.18i)11-s + (−1.27 − 1.17i)12-s + (3.59 + 0.247i)13-s + (0.596 − 2.57i)14-s + (0.0978 + 0.312i)15-s + (−0.5 + 0.866i)16-s + (−2.16 − 3.74i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.954 − 0.298i)3-s + (−0.249 − 0.433i)4-s + 0.0844i·5-s + (−0.154 + 0.690i)6-s + (−0.956 + 0.291i)7-s + 0.353·8-s + (0.821 − 0.570i)9-s + (−0.0517 − 0.0298i)10-s + (0.902 − 1.56i)11-s + (−0.368 − 0.338i)12-s + (0.997 + 0.0687i)13-s + (0.159 − 0.688i)14-s + (0.0252 + 0.0806i)15-s + (−0.125 + 0.216i)16-s + (−0.524 − 0.908i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0203i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.62402 + 0.0165000i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.62402 + 0.0165000i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-1.65 + 0.517i)T \) |
| 7 | \( 1 + (2.53 - 0.772i)T \) |
| 13 | \( 1 + (-3.59 - 0.247i)T \) |
good | 5 | \( 1 - 0.188iT - 5T^{2} \) |
| 11 | \( 1 + (-2.99 + 5.18i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (2.16 + 3.74i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.70 - 4.69i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.12 - 1.22i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.72 - 3.30i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 7.73T + 31T^{2} \) |
| 37 | \( 1 + (-1.21 - 0.699i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (7.23 + 4.17i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.00 + 3.48i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 3.00iT - 47T^{2} \) |
| 53 | \( 1 - 0.440iT - 53T^{2} \) |
| 59 | \( 1 + (1.61 - 0.929i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.69 + 0.976i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.40 - 5.42i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.71 + 2.96i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 7.64T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 - 16.1iT - 83T^{2} \) |
| 89 | \( 1 + (12.3 + 7.14i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.15 - 14.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.60625984831673472631943449947, −9.527394562903457593107332510416, −8.840386809824349954975274017116, −8.429007872930789912961274703947, −7.08796577261497564621035745544, −6.51538488607666595538305238768, −5.54217370493414155735852282892, −3.77261379806900858181024064301, −3.05829323261477453845431288334, −1.15736664861058020603074921413,
1.52887789952197336063322128381, 2.88094584365151301933910886218, 3.87130391182858188767868001987, 4.68970929376349700358537006822, 6.59621660586247790630933490253, 7.23543148754917316685719287641, 8.524826142398766943298136747493, 9.124050813962091853146437511230, 9.845709576564622529687726402877, 10.52979166837540565164686929569