L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.499 − 0.866i)6-s + (3.08 − 1.78i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (0.280 − 0.486i)11-s − 0.999i·12-s + (−2.21 + 2.84i)13-s + 3.56·14-s + (−0.5 + 0.866i)16-s + (2.70 − 1.56i)17-s + 0.999i·18-s + (−1.21 − 2.11i)19-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.204 − 0.353i)6-s + (1.16 − 0.673i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.0846 − 0.146i)11-s − 0.288i·12-s + (−0.615 + 0.788i)13-s + 0.951·14-s + (−0.125 + 0.216i)16-s + (0.655 − 0.378i)17-s + 0.235i·18-s + (−0.279 − 0.484i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.499250176\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.499250176\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (2.21 - 2.84i)T \) |
good | 7 | \( 1 + (-3.08 + 1.78i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.280 + 0.486i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.70 + 1.56i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.21 + 2.11i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.65 - 3.84i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.561 + 0.972i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + (-0.486 - 0.280i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.56 + 2.70i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.379 + 0.219i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 4.24iT - 53T^{2} \) |
| 59 | \( 1 + (5.12 + 8.87i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.842 - 1.45i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.2 - 5.90i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (6.28 + 10.8i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 9iT - 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 - 6.56iT - 83T^{2} \) |
| 89 | \( 1 + (-5.12 + 8.87i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.43 + 1.40i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.106511140166917089664794234095, −8.168798121874382924291134627247, −7.37512221095938026249809653851, −6.97981218589874513385458279362, −5.98075478374452601984095217024, −4.94539475322215549227929268519, −4.68807472968300394945421048440, −3.53108375555297639513992927369, −2.24173247857248342261338654467, −1.03912591936287826653756121260,
1.07978222182576244529399088228, 2.29663838074397059155303100401, 3.28376177619596730221954981181, 4.50609049414155011970068082073, 5.02286278752717122124006768657, 5.70936909058151750231159262735, 6.55960906056104492394418461866, 7.64143348876538699891685933051, 8.346025818490029999477252713953, 9.270602105065159746983948014626