L(s) = 1 | + (1.36 − 0.366i)2-s + (−1.66 + 2.49i)3-s + (1.73 − i)4-s + (1.09 − 4.87i)5-s + (−1.36 + 4.01i)6-s + (−1.72 − 6.78i)7-s + (1.99 − 2i)8-s + (−3.43 − 8.31i)9-s + (−0.293 − 7.06i)10-s + (5.25 − 3.03i)11-s + (−0.394 + 5.98i)12-s + (4.95 − 4.95i)13-s + (−4.83 − 8.63i)14-s + (10.3 + 10.8i)15-s + (1.99 − 3.46i)16-s + (24.7 + 6.62i)17-s + ⋯ |
L(s) = 1 | + (0.683 − 0.183i)2-s + (−0.555 + 0.831i)3-s + (0.433 − 0.250i)4-s + (0.218 − 0.975i)5-s + (−0.227 + 0.669i)6-s + (−0.246 − 0.969i)7-s + (0.249 − 0.250i)8-s + (−0.381 − 0.924i)9-s + (−0.0293 − 0.706i)10-s + (0.477 − 0.275i)11-s + (−0.0328 + 0.498i)12-s + (0.381 − 0.381i)13-s + (−0.345 − 0.616i)14-s + (0.689 + 0.724i)15-s + (0.124 − 0.216i)16-s + (1.45 + 0.389i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.68026 - 0.895693i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.68026 - 0.895693i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.36 + 0.366i)T \) |
| 3 | \( 1 + (1.66 - 2.49i)T \) |
| 5 | \( 1 + (-1.09 + 4.87i)T \) |
| 7 | \( 1 + (1.72 + 6.78i)T \) |
good | 11 | \( 1 + (-5.25 + 3.03i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-4.95 + 4.95i)T - 169iT^{2} \) |
| 17 | \( 1 + (-24.7 - 6.62i)T + (250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (8.47 - 14.6i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (7.44 + 27.7i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 - 29.7T + 841T^{2} \) |
| 31 | \( 1 + (36.5 - 21.1i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (12.4 + 46.2i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + 2.99T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-22.2 - 22.2i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-16.3 - 61.1i)T + (-1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-1.70 - 0.456i)T + (2.43e3 + 1.40e3i)T^{2} \) |
| 59 | \( 1 + (72.6 - 41.9i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (18.0 + 10.4i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-32.1 - 8.61i)T + (3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 - 10.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-94.5 - 25.3i)T + (4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-74.6 - 43.1i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-47.0 - 47.0i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (-138. - 80.0i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (81.7 + 81.7i)T + 9.40e3iT^{2} \) |
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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
−12.34731133426133965180656792368, −10.87803028690326942229871391701, −10.31052259718017102119393782317, −9.283923674397646560420470509944, −8.021402777839667878354631108042, −6.37702991462072272905549145628, −5.53767159609364471217887323261, −4.36673025054316244894242524374, −3.55433747312490104943030613899, −0.998832820137927859811652397147,
2.00562477443287170428804462285, 3.32485772666970293992205407020, 5.23179463195865922307252346132, 6.13210441777554733087537826083, 6.89121841134561568866645681563, 7.899170428918328057070957830810, 9.385018280178760049088899128291, 10.66994226782581281830300476660, 11.74166321673577384569212441181, 12.10159696634673086475120515193