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 + (−2.73 − 1.58i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−1.5 − 2.59i)11-s + 0.999i·12-s + (−1.87 − 3.08i)13-s + 3.16·14-s + (−0.5 − 0.866i)16-s + (6.20 + 3.58i)17-s + 0.999i·18-s + (1.58 − 2.73i)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 + (−1.03 − 0.597i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.452 − 0.783i)11-s + 0.288i·12-s + (−0.519 − 0.854i)13-s + 0.845·14-s + (−0.125 − 0.216i)16-s + (1.50 + 0.868i)17-s + 0.235i·18-s + (0.362 − 0.628i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.07468793489\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07468793489\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.87 + 3.08i)T \) |
good | 7 | \( 1 + (2.73 + 1.58i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-6.20 - 3.58i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.58 + 2.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.60 - 2.08i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.16 + 7.20i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 9.16T + 31T^{2} \) |
| 37 | \( 1 + (9.08 - 5.24i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.74 - 8.21i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.47 + 2.58i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 7.16iT - 53T^{2} \) |
| 59 | \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7.24 - 12.5i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.46 - 2i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.91 + 6.78i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 1.32iT - 73T^{2} \) |
| 79 | \( 1 + 9.16T + 79T^{2} \) |
| 83 | \( 1 + 13.6iT - 83T^{2} \) |
| 89 | \( 1 + (3.58 + 6.20i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.04 - 2.33i)T + (48.5 + 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.925859125051328948581054775395, −8.733692183430738623221564479834, −7.86715637353843204996564337490, −7.35353668223320396445201349476, −6.17396232640449795012452597444, −5.88581365115436164920982319106, −4.87196597150128535196280142884, −3.63182615586889566831606207354, −2.87202269873938630015122116637, −1.05863154599423683543359176453,
0.04144562858573414914770073029, 1.62948845732596183606672771204, 2.65777626449106021078551340781, 3.59205678070229762975642628734, 4.88436785085206757710602813783, 5.68470571534599844075669584581, 6.63915013123682400941157702680, 7.27091088488216361008704525742, 7.991503282412463660375754854637, 9.039083189573141604416614928904