| L(s) = 1 | + (9.99 + 17.3i)5-s + 36.5·7-s + 119.·11-s + (−124. − 71.9i)13-s + (−262. − 454. i)17-s + (341. + 115. i)19-s + (111. − 192. i)23-s + (112. − 195. i)25-s + (−812. − 469. i)29-s − 715. i·31-s + (365. + 633. i)35-s − 1.84e3i·37-s + (−238. + 137. i)41-s + (886. + 1.53e3i)43-s + (−335. + 581. i)47-s + ⋯ |
| L(s) = 1 | + (0.399 + 0.692i)5-s + 0.746·7-s + 0.985·11-s + (−0.737 − 0.425i)13-s + (−0.907 − 1.57i)17-s + (0.947 + 0.320i)19-s + (0.210 − 0.364i)23-s + (0.180 − 0.312i)25-s + (−0.966 − 0.558i)29-s − 0.744i·31-s + (0.298 + 0.517i)35-s − 1.34i·37-s + (−0.141 + 0.0817i)41-s + (0.479 + 0.830i)43-s + (−0.151 + 0.263i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 + 0.812i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.583 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.284964521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.284964521\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-341. - 115. i)T \) |
| good | 5 | \( 1 + (-9.99 - 17.3i)T + (-312.5 + 541. i)T^{2} \) |
| 7 | \( 1 - 36.5T + 2.40e3T^{2} \) |
| 11 | \( 1 - 119.T + 1.46e4T^{2} \) |
| 13 | \( 1 + (124. + 71.9i)T + (1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 + (262. + 454. i)T + (-4.17e4 + 7.23e4i)T^{2} \) |
| 23 | \( 1 + (-111. + 192. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (812. + 469. i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + 715. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.84e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (238. - 137. i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-886. - 1.53e3i)T + (-1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (335. - 581. i)T + (-2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + (-604. - 349. i)T + (3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (80.5 - 46.5i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (481. - 833. i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (1.33e3 + 769. i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (-2.55e3 + 1.47e3i)T + (1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + (1.18e3 + 2.05e3i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-9.51e3 + 5.49e3i)T + (1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + 9.51e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + (-9.88e3 - 5.70e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + (-6.99e3 + 4.04e3i)T + (4.42e7 - 7.66e7i)T^{2} \) |
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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
−9.611309972501317882578276897165, −9.166828160659040829215996141991, −7.83338104387344389905733371783, −7.18403296985976720537590397959, −6.25647392324078288351978038593, −5.20843075841085417059542182231, −4.27728433468253643309785456402, −2.93916932539152432655225090707, −2.00657050079570006518755415807, −0.56178760691929651407135571782,
1.22391454904668162431420335044, 1.93209472853860687913771043587, 3.57550848748712146621107308258, 4.64587156011138585633722424308, 5.36321556700870120230623146949, 6.50370253016243341631286781815, 7.37687552260532346392647934363, 8.524239591773680582068073867451, 9.055985390252856740060435229216, 9.890794454165134621146434075075
