| L(s) = 1 | + (1.86 + 1.23i)5-s − 0.746i·7-s − 5.36i·11-s − 2.92·13-s − 2.13i·17-s − 1.73i·19-s − 7.49i·23-s + (1.92 + 4.61i)25-s − 6.74i·29-s − 2.64·31-s + (0.925 − 1.38i)35-s − 1.07·37-s + 11.2·41-s − 7.44·43-s + 1.73i·47-s + ⋯ |
| L(s) = 1 | + (0.832 + 0.554i)5-s − 0.282i·7-s − 1.61i·11-s − 0.811·13-s − 0.517i·17-s − 0.397i·19-s − 1.56i·23-s + (0.385 + 0.922i)25-s − 1.25i·29-s − 0.475·31-s + (0.156 − 0.234i)35-s − 0.176·37-s + 1.76·41-s − 1.13·43-s + 0.252i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.653025560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.653025560\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.86 - 1.23i)T \) |
| good | 7 | \( 1 + 0.746iT - 7T^{2} \) |
| 11 | \( 1 + 5.36iT - 11T^{2} \) |
| 13 | \( 1 + 2.92T + 13T^{2} \) |
| 17 | \( 1 + 2.13iT - 17T^{2} \) |
| 19 | \( 1 + 1.73iT - 19T^{2} \) |
| 23 | \( 1 + 7.49iT - 23T^{2} \) |
| 29 | \( 1 + 6.74iT - 29T^{2} \) |
| 31 | \( 1 + 2.64T + 31T^{2} \) |
| 37 | \( 1 + 1.07T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 7.44T + 43T^{2} \) |
| 47 | \( 1 - 1.73iT - 47T^{2} \) |
| 53 | \( 1 + 7.72T + 53T^{2} \) |
| 59 | \( 1 - 6.85iT - 59T^{2} \) |
| 61 | \( 1 - 6.45iT - 61T^{2} \) |
| 67 | \( 1 - 7.44T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 - 0.690iT - 73T^{2} \) |
| 79 | \( 1 - 2.64T + 79T^{2} \) |
| 83 | \( 1 + 5.85T + 83T^{2} \) |
| 89 | \( 1 - 7.59T + 89T^{2} \) |
| 97 | \( 1 + 14.1iT - 97T^{2} \) |
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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
−9.423228750785570233719715049265, −8.660933133060370585081882790909, −7.74286462612622743521469499372, −6.82672722508742532412657688726, −6.12113906088393951399502042289, −5.36231635820950721486461726058, −4.27779972281684461919115233420, −3.04590940349524161916185500591, −2.34532674122059205921360451456, −0.65635533558555377851112369739,
1.55838631837197677616013559892, 2.29565540139803895615350775928, 3.72292975705491023808755920755, 4.90084037144086778086310483785, 5.33936244147344951200965525702, 6.41435981105441820893557054241, 7.27805225283611125989126819398, 8.036350565127538139882277348067, 9.174707535407614244715053969809, 9.580938275058193985757239193380
