L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.866 − 1.5i)3-s + (−0.499 − 0.866i)4-s − 1.73·6-s + (0.866 − 2.5i)7-s − 0.999·8-s + (−1.5 + 2.59i)9-s + (−5.19 + 3i)11-s + (−0.866 + 1.49i)12-s − 1.73·13-s + (−1.73 − 2i)14-s + (−0.5 + 0.866i)16-s + (1.5 + 2.59i)18-s + (1.5 + 0.866i)19-s + (−4.5 + 0.866i)21-s + 6i·22-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.499 − 0.866i)3-s + (−0.249 − 0.433i)4-s − 0.707·6-s + (0.327 − 0.944i)7-s − 0.353·8-s + (−0.5 + 0.866i)9-s + (−1.56 + 0.904i)11-s + (−0.250 + 0.433i)12-s − 0.480·13-s + (−0.462 − 0.534i)14-s + (−0.125 + 0.216i)16-s + (0.353 + 0.612i)18-s + (0.344 + 0.198i)19-s + (−0.981 + 0.188i)21-s + 1.27i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00342 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00342 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1247070852\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1247070852\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.866 + 1.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.866 + 2.5i)T \) |
good | 11 | \( 1 + (5.19 - 3i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 1.73T + 13T^{2} \) |
| 17 | \( 1 + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.5 - 0.866i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + (3 - 1.73i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.06 + 3.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + (-9 - 5.19i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.19 - 9i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.5 + 2.59i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.52 + 5.5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 + (-0.866 - 1.5i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 10.3iT - 83T^{2} \) |
| 89 | \( 1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−9.632591899953686636402888601803, −8.182054283068707958927934698779, −7.53509403724964171965183733009, −6.92440851694550517234406417053, −5.55818140196312636944597184304, −5.07009957790311723308358490521, −3.94625562668293400932370992013, −2.55183335024210688193218098985, −1.61408800900138238871413863659, −0.05025182912258801397977523048,
2.57053463568489255891335335761, 3.50907079299549538759904424829, 4.88637166051936037371148203168, 5.27412200061023724039321705914, 6.00525852159153699691452520646, 7.05092984313535325584939038244, 8.301972449046399740839218014797, 8.627721037396605431600703437280, 9.730845982779835143936387346047, 10.53429918243606038748506591062