| 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
−9.890794454165134621146434075075, −9.055985390252856740060435229216, −8.524239591773680582068073867451, −7.37687552260532346392647934363, −6.50370253016243341631286781815, −5.36321556700870120230623146949, −4.64587156011138585633722424308, −3.57550848748712146621107308258, −1.93209472853860687913771043587, −1.22391454904668162431420335044,
0.56178760691929651407135571782, 2.00657050079570006518755415807, 2.93916932539152432655225090707, 4.27728433468253643309785456402, 5.20843075841085417059542182231, 6.25647392324078288351978038593, 7.18403296985976720537590397959, 7.83338104387344389905733371783, 9.166828160659040829215996141991, 9.611309972501317882578276897165
