| L(s) = 1 | + (−1.91 + 0.511i)2-s + (−1.67 − 0.448i)3-s + (−0.0771 + 0.0445i)4-s + (−4.99 − 0.258i)5-s + 3.42·6-s + (6.99 + 0.240i)7-s + (5.71 − 5.71i)8-s + (2.59 + 1.50i)9-s + (9.67 − 2.06i)10-s + (1.58 + 2.74i)11-s + (0.148 − 0.0399i)12-s + (11.0 − 11.0i)13-s + (−13.4 + 3.12i)14-s + (8.23 + 2.67i)15-s + (−7.81 + 13.5i)16-s + (4.27 − 15.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.955 + 0.255i)2-s + (−0.557 − 0.149i)3-s + (−0.0192 + 0.0111i)4-s + (−0.998 − 0.0516i)5-s + 0.570·6-s + (0.999 + 0.0343i)7-s + (0.714 − 0.714i)8-s + (0.288 + 0.166i)9-s + (0.967 − 0.206i)10-s + (0.144 + 0.249i)11-s + (0.0124 − 0.00332i)12-s + (0.850 − 0.850i)13-s + (−0.963 + 0.222i)14-s + (0.549 + 0.178i)15-s + (−0.488 + 0.846i)16-s + (0.251 − 0.938i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.598360 - 0.0765371i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.598360 - 0.0765371i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.67 + 0.448i)T \) |
| 5 | \( 1 + (4.99 + 0.258i)T \) |
| 7 | \( 1 + (-6.99 - 0.240i)T \) |
| good | 2 | \( 1 + (1.91 - 0.511i)T + (3.46 - 2i)T^{2} \) |
| 11 | \( 1 + (-1.58 - 2.74i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-11.0 + 11.0i)T - 169iT^{2} \) |
| 17 | \( 1 + (-4.27 + 15.9i)T + (-250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (-24.9 - 14.4i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (2.86 + 10.6i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 - 20.9iT - 841T^{2} \) |
| 31 | \( 1 + (30.4 + 52.7i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (4.96 - 1.32i)T + (1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 - 0.605T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-16.0 + 16.0i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-34.1 + 9.14i)T + (1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-60.8 - 16.2i)T + (2.43e3 + 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-88.6 + 51.2i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (16.9 - 29.4i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (13.4 - 50.2i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 25.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + (86.1 + 23.0i)T + (4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-6.66 - 3.84i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-81.1 + 81.1i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (35.6 + 20.6i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-1.74 - 1.74i)T + 9.40e3iT^{2} \) |
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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
−13.38768057842872499441289390785, −12.17240951562034646315866087902, −11.30285531251863292419105838534, −10.30395281396854237853151493169, −8.931095645719484493751849973829, −7.86865209863193006638726989752, −7.29836575194214987572603341791, −5.37025772323663401867529340671, −3.94606502643475957377560698956, −0.883088260058526439860057423959,
1.22132687042244611180142399003, 4.03689329095051831787445493645, 5.32208963275622852982254308051, 7.19179375973985874536531880500, 8.272275259118492804886428131745, 9.142303784320880469224119978044, 10.60088735612720213836580379644, 11.25797172733977314017374081772, 11.97154213859831927800287680464, 13.64907682315353595883154078061
