L(s) = 1 | + (−0.620 − 1.90i)2-s + (1.67 − 0.448i)3-s + (−3.23 + 2.35i)4-s + (−3.50 − 0.939i)5-s + (−1.89 − 2.90i)6-s + (−1.65 + 6.80i)7-s + (6.48 + 4.67i)8-s + (2.59 − 1.50i)9-s + (0.388 + 7.24i)10-s + (14.5 − 3.91i)11-s + (−4.34 + 5.39i)12-s + (0.646 + 0.646i)13-s + (13.9 − 1.07i)14-s − 6.28·15-s + (4.87 − 15.2i)16-s + (17.0 + 9.84i)17-s + ⋯ |
L(s) = 1 | + (−0.310 − 0.950i)2-s + (0.557 − 0.149i)3-s + (−0.807 + 0.589i)4-s + (−0.700 − 0.187i)5-s + (−0.315 − 0.483i)6-s + (−0.236 + 0.971i)7-s + (0.811 + 0.584i)8-s + (0.288 − 0.166i)9-s + (0.0388 + 0.724i)10-s + (1.32 − 0.355i)11-s + (−0.362 + 0.449i)12-s + (0.0497 + 0.0497i)13-s + (0.997 − 0.0765i)14-s − 0.418·15-s + (0.304 − 0.952i)16-s + (1.00 + 0.578i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.474 + 0.880i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.28750 - 0.768184i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28750 - 0.768184i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.620 + 1.90i)T \) |
| 3 | \( 1 + (-1.67 + 0.448i)T \) |
| 7 | \( 1 + (1.65 - 6.80i)T \) |
good | 5 | \( 1 + (3.50 + 0.939i)T + (21.6 + 12.5i)T^{2} \) |
| 11 | \( 1 + (-14.5 + 3.91i)T + (104. - 60.5i)T^{2} \) |
| 13 | \( 1 + (-0.646 - 0.646i)T + 169iT^{2} \) |
| 17 | \( 1 + (-17.0 - 9.84i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-8.72 + 32.5i)T + (-312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (-3.45 + 1.99i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-7.69 + 7.69i)T - 841iT^{2} \) |
| 31 | \( 1 + (-44.7 - 25.8i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (8.90 - 33.2i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 - 57.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-4.23 - 4.23i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (50.4 - 29.1i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-83.4 + 22.3i)T + (2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-25.7 - 96.1i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (-22.0 + 82.2i)T + (-3.22e3 - 1.86e3i)T^{2} \) |
| 67 | \( 1 + (16.0 + 59.9i)T + (-3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + 60.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (4.60 - 7.97i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (16.6 + 28.8i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (64.4 + 64.4i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (-26.8 - 46.4i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 13.9iT - 9.40e3T^{2} \) |
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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
−11.55162060478520610074658179346, −10.14892843261628789262358747905, −9.184532820633737889879844478396, −8.655192391739527658619031940216, −7.76637204569547945289680051912, −6.41151091575728808760773648161, −4.80115887711904052938350838894, −3.64606900161259334742628496989, −2.68761020493155396322279741187, −1.05128427942739192883842365923,
1.10173930113693993360907511886, 3.64523452879223188871344184850, 4.23591084486396239077005376409, 5.79933977967860482833551830334, 7.02884481584228714773981149917, 7.60069468949667284590758222723, 8.459706278363592873424474757776, 9.718465264481023992459203224284, 10.04603775925041262612833407958, 11.44736123087813842568056981338