| L(s) = 1 | + 0.585·5-s + 5.15·7-s + 11.8·11-s − 14.9i·13-s − 8.91·17-s + (−18.4 − 4.41i)19-s − 35.6·23-s − 24.6·25-s − 7.12i·29-s − 10.8i·31-s + 3.02·35-s + 51.3i·37-s + 33.2i·41-s − 54.2·43-s − 32.6·47-s + ⋯ |
| L(s) = 1 | + 0.117·5-s + 0.737·7-s + 1.07·11-s − 1.14i·13-s − 0.524·17-s + (−0.972 − 0.232i)19-s − 1.55·23-s − 0.986·25-s − 0.245i·29-s − 0.348i·31-s + 0.0863·35-s + 1.38i·37-s + 0.810i·41-s − 1.26·43-s − 0.694·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.232i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.972 - 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.02631834632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02631834632\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (18.4 + 4.41i)T \) |
| good | 5 | \( 1 - 0.585T + 25T^{2} \) |
| 7 | \( 1 - 5.15T + 49T^{2} \) |
| 11 | \( 1 - 11.8T + 121T^{2} \) |
| 13 | \( 1 + 14.9iT - 169T^{2} \) |
| 17 | \( 1 + 8.91T + 289T^{2} \) |
| 23 | \( 1 + 35.6T + 529T^{2} \) |
| 29 | \( 1 + 7.12iT - 841T^{2} \) |
| 31 | \( 1 + 10.8iT - 961T^{2} \) |
| 37 | \( 1 - 51.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 33.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 54.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 32.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 91.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 107. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 23.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 107. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 72.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.25T + 5.32e3T^{2} \) |
| 79 | \( 1 - 45.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 132.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 42.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 143. iT - 9.40e3T^{2} \) |
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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
−8.893445697114682848229121403142, −8.140384391451634379672997037900, −7.74857962673485628279845849941, −6.46451289837144916584326263614, −6.14482240705374362131042941674, −5.02295470978238013340341904039, −4.31073265381738799246012908485, −3.45304924239280125747280283450, −2.24609837905451168828820548119, −1.38942788513993473551935742028,
0.00553327715345086477110533393, 1.70484825712870040870977998932, 2.06171954276976086514149482992, 3.75236283434337405671026983583, 4.18836430123310532562569191428, 5.10923810211602875344146293531, 6.18292511626590267175004398714, 6.62906759942337335430909783805, 7.57950391282363913673759312853, 8.391997831982167677821015417286
