L(s) = 1 | + (1.22 − 1.22i)3-s + (2 − 2i)5-s + 5.27·7-s − 2.99i·9-s + (0.757 + 0.757i)11-s + (0.464 + 0.464i)13-s − 4.89i·15-s + 15.8·17-s + (3.96 − 3.96i)19-s + (6.46 − 6.46i)21-s + 21.8·23-s + 17i·25-s + (−3.67 − 3.67i)27-s + (−14.9 − 14.9i)29-s − 57.2i·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (0.400 − 0.400i)5-s + 0.753·7-s − 0.333i·9-s + (0.0688 + 0.0688i)11-s + (0.0357 + 0.0357i)13-s − 0.326i·15-s + 0.932·17-s + (0.208 − 0.208i)19-s + (0.307 − 0.307i)21-s + 0.950·23-s + 0.680i·25-s + (−0.136 − 0.136i)27-s + (−0.514 − 0.514i)29-s − 1.84i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.622785741\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.622785741\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.22 + 1.22i)T \) |
good | 5 | \( 1 + (-2 + 2i)T - 25iT^{2} \) |
| 7 | \( 1 - 5.27T + 49T^{2} \) |
| 11 | \( 1 + (-0.757 - 0.757i)T + 121iT^{2} \) |
| 13 | \( 1 + (-0.464 - 0.464i)T + 169iT^{2} \) |
| 17 | \( 1 - 15.8T + 289T^{2} \) |
| 19 | \( 1 + (-3.96 + 3.96i)T - 361iT^{2} \) |
| 23 | \( 1 - 21.8T + 529T^{2} \) |
| 29 | \( 1 + (14.9 + 14.9i)T + 841iT^{2} \) |
| 31 | \( 1 + 57.2iT - 961T^{2} \) |
| 37 | \( 1 + (-14.6 + 14.6i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 79.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-15.2 - 15.2i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + 2.27iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-27.2 + 27.2i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (50.5 + 50.5i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (68.8 + 68.8i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (-59.9 + 59.9i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 82.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 77.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 18.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-99.9 + 99.9i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 74iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 49.2T + 9.40e3T^{2} \) |
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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
−9.624400886887265057595810485244, −9.336385847121492941847537020646, −8.050437086800832325860689279391, −7.71411740959495297418191324931, −6.49541114270781461390311156104, −5.51605193387354128692060683749, −4.62189664600405221506597404729, −3.34931474063031249688657569236, −2.07483613610270189873227951915, −1.00081394155900860351942044083,
1.36448616631376978180067425169, 2.67974233334581201677271742523, 3.68425753860444171810621998519, 4.89179203346632771113095946109, 5.65073690828279528417281364093, 6.87416632688915552206143069176, 7.71517710697181419467996526894, 8.638121060334884085270452526517, 9.334357480584404565863794087812, 10.44559239488540446627385624110