| L(s) = 1 | + (0.299 − 0.269i)2-s + (−1.24 + 1.20i)3-s + (−0.192 + 1.82i)4-s + (−1.79 − 1.32i)5-s + (−0.0494 + 0.696i)6-s + (−4.03 − 2.33i)7-s + (0.909 + 1.25i)8-s + (0.110 − 2.99i)9-s + (−0.897 + 0.0865i)10-s + (1.78 + 1.98i)11-s + (−1.95 − 2.50i)12-s + (−3.63 − 3.27i)13-s + (−1.83 + 0.390i)14-s + (3.84 − 0.502i)15-s + (−2.98 − 0.634i)16-s + (0.388 + 0.534i)17-s + ⋯ |
| L(s) = 1 | + (0.211 − 0.190i)2-s + (−0.720 + 0.693i)3-s + (−0.0960 + 0.913i)4-s + (−0.803 − 0.594i)5-s + (−0.0201 + 0.284i)6-s + (−1.52 − 0.881i)7-s + (0.321 + 0.442i)8-s + (0.0369 − 0.999i)9-s + (−0.283 + 0.0273i)10-s + (0.539 + 0.598i)11-s + (−0.564 − 0.724i)12-s + (−1.00 − 0.907i)13-s + (−0.491 + 0.104i)14-s + (0.991 − 0.129i)15-s + (−0.746 − 0.158i)16-s + (0.0941 + 0.129i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00840343 - 0.0720735i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00840343 - 0.0720735i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.24 - 1.20i)T \) |
| 5 | \( 1 + (1.79 + 1.32i)T \) |
| good | 2 | \( 1 + (-0.299 + 0.269i)T + (0.209 - 1.98i)T^{2} \) |
| 7 | \( 1 + (4.03 + 2.33i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.78 - 1.98i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (3.63 + 3.27i)T + (1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.388 - 0.534i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (3.29 - 2.39i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.898 - 4.22i)T + (-21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (8.03 - 3.57i)T + (19.4 - 21.5i)T^{2} \) |
| 31 | \( 1 + (-1.01 - 0.450i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (3.15 - 1.02i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-4.70 + 5.22i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (1.95 + 1.12i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.656 + 1.47i)T + (-31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-4.22 + 5.81i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-3.58 + 3.97i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (6.67 + 7.40i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-1.07 + 2.42i)T + (-44.8 - 49.7i)T^{2} \) |
| 71 | \( 1 + (-8.66 - 6.29i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (9.14 + 2.97i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (3.11 - 1.38i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (1.35 - 0.142i)T + (81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (-1.57 + 4.84i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-6.03 - 13.5i)T + (-64.9 + 72.0i)T^{2} \) |
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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
−12.58968143089729148394572566377, −12.05040153964392348599628654476, −10.90768426706859252531639120142, −9.905271659039265023981014652673, −9.074048978233838375321378253083, −7.62986161524530522008865814152, −6.82888296111430549761444411826, −5.22130731341437450809451237788, −3.99058902398587740379610343658, −3.46971468815590052670845863703,
0.05802338300569010126502408821, 2.53069520785612802548057210190, 4.35506712007234172474719662497, 5.82641993990440239928789734013, 6.52057458648684576490144512010, 7.19157203725596528605465362330, 8.879746693481902509937940052429, 9.861162392573876646086944433349, 10.95148725152746361136094084952, 11.77945325511013583412126926660
