L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + 0.267·5-s + (−0.499 − 0.866i)6-s + (−0.366 − 0.633i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.133 + 0.232i)10-s + (−2.36 + 4.09i)11-s + 0.999·12-s + 0.732·14-s + (−0.133 + 0.232i)15-s + (−0.5 + 0.866i)16-s + (1.13 + 1.96i)17-s + 0.999·18-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + 0.119·5-s + (−0.204 − 0.353i)6-s + (−0.138 − 0.239i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.0423 + 0.0733i)10-s + (−0.713 + 1.23i)11-s + 0.288·12-s + 0.195·14-s + (−0.0345 + 0.0599i)15-s + (−0.125 + 0.216i)16-s + (0.275 + 0.476i)17-s + 0.235·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2544194859\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2544194859\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 0.267T + 5T^{2} \) |
| 7 | \( 1 + (0.366 + 0.633i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.36 - 4.09i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.13 - 1.96i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.633 + 1.09i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.09 + 5.36i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.23 - 2.13i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 + (5.23 - 9.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.69 - 9.86i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.83 + 6.63i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 8.19T + 47T^{2} \) |
| 53 | \( 1 - 0.464T + 53T^{2} \) |
| 59 | \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.598 + 1.03i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.56 + 9.63i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.633 + 1.09i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 9.73T + 73T^{2} \) |
| 79 | \( 1 + 9.46T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + (-1.26 + 2.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3 - 5.19i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.15817769976471034641876823697, −9.833933204662265285326723351906, −8.782290185763782532697803376238, −7.986356771322807808597439334122, −7.03828032221575775008672475905, −6.36687108552210235542888378234, −5.18967739445822727903694396601, −4.67168461357426992200885985600, −3.40568803763984963013532894067, −1.84334037570449778381363369033,
0.13119698711258467608477557001, 1.64468497114695402280470238372, 2.85063198901189601756951461318, 3.77740580666081781485826414909, 5.33992941744246851468231783403, 5.81282787227736384873495479629, 7.12237749825226157651942607438, 7.82911215963849597612330752958, 8.704895326982031776642335278624, 9.446754883144790682472546950820