L(s) = 1 | + (−0.309 + 0.951i)3-s + (1.30 − 0.951i)5-s + (−0.0729 − 0.224i)7-s + (−0.809 − 0.587i)9-s + (0.809 + 3.21i)11-s + (3.42 + 2.48i)13-s + (0.499 + 1.53i)15-s + (0.118 − 0.0857i)17-s + (2.11 − 6.51i)19-s + 0.236·21-s + 5·23-s + (−0.736 + 2.26i)25-s + (0.809 − 0.587i)27-s + (0.618 + 1.90i)29-s + (2.73 + 1.98i)31-s + ⋯ |
L(s) = 1 | + (−0.178 + 0.549i)3-s + (0.585 − 0.425i)5-s + (−0.0275 − 0.0848i)7-s + (−0.269 − 0.195i)9-s + (0.243 + 0.969i)11-s + (0.950 + 0.690i)13-s + (0.129 + 0.397i)15-s + (0.0286 − 0.0207i)17-s + (0.485 − 1.49i)19-s + 0.0515·21-s + 1.04·23-s + (−0.147 + 0.453i)25-s + (0.155 − 0.113i)27-s + (0.114 + 0.353i)29-s + (0.491 + 0.357i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.49662 + 0.450156i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.49662 + 0.450156i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 + (-0.809 - 3.21i)T \) |
good | 5 | \( 1 + (-1.30 + 0.951i)T + (1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (0.0729 + 0.224i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (-3.42 - 2.48i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.118 + 0.0857i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.11 + 6.51i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 - 5T + 23T^{2} \) |
| 29 | \( 1 + (-0.618 - 1.90i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.73 - 1.98i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.690 - 2.12i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (2.69 - 8.28i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 2.52T + 43T^{2} \) |
| 47 | \( 1 + (2.95 - 9.09i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (5.35 + 3.88i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (3.64 + 11.2i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-8.78 + 6.37i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 4.38T + 67T^{2} \) |
| 71 | \( 1 + (11.7 - 8.55i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (4.85 + 14.9i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (7.04 + 5.11i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (1.42 - 1.03i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 5.18T + 89T^{2} \) |
| 97 | \( 1 + (12.1 + 8.83i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.03680299065074745378352963401, −9.834709219316129492509996481986, −9.327461875385622003763270710195, −8.548634553396841733906651259541, −7.14671588223886397448961342554, −6.35253735304241426841730641149, −5.11680339831629015324186286983, −4.47235306557935738485096643253, −3.07802707238408678609122230808, −1.44583843192524695666275196921,
1.16481504670238952861393537666, 2.70706353928177007759572660854, 3.82820204515274875978879110040, 5.64098103540692299657699111564, 5.96934643472988870647415420071, 7.06121320135829142292749847952, 8.131299794303231184660347493953, 8.847206689438132349695702323911, 10.08124489331243158531221118454, 10.73547275145947364181234771049