| L(s) = 1 | + 1.73i·5-s + 4.73·7-s − 4.73i·11-s − 3i·13-s + 7.73·17-s + 6.19i·19-s − 4.73·23-s + 2.00·25-s + 7.73i·29-s + 3.46·31-s + 8.19i·35-s + 6.46i·37-s − 3.46·41-s + 4.19i·43-s − 0.928·47-s + ⋯ |
| L(s) = 1 | + 0.774i·5-s + 1.78·7-s − 1.42i·11-s − 0.832i·13-s + 1.87·17-s + 1.42i·19-s − 0.986·23-s + 0.400·25-s + 1.43i·29-s + 0.622·31-s + 1.38i·35-s + 1.06i·37-s − 0.541·41-s + 0.639i·43-s − 0.135·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.755875582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.755875582\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 7 | \( 1 - 4.73T + 7T^{2} \) |
| 11 | \( 1 + 4.73iT - 11T^{2} \) |
| 13 | \( 1 + 3iT - 13T^{2} \) |
| 17 | \( 1 - 7.73T + 17T^{2} \) |
| 19 | \( 1 - 6.19iT - 19T^{2} \) |
| 23 | \( 1 + 4.73T + 23T^{2} \) |
| 29 | \( 1 - 7.73iT - 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 - 6.46iT - 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 - 4.19iT - 43T^{2} \) |
| 47 | \( 1 + 0.928T + 47T^{2} \) |
| 53 | \( 1 + 12.9iT - 53T^{2} \) |
| 59 | \( 1 + 2.53iT - 59T^{2} \) |
| 61 | \( 1 - 0.464iT - 61T^{2} \) |
| 67 | \( 1 + 6.19iT - 67T^{2} \) |
| 71 | \( 1 + 1.26T + 71T^{2} \) |
| 73 | \( 1 - 5T + 73T^{2} \) |
| 79 | \( 1 + 3.80T + 79T^{2} \) |
| 83 | \( 1 + 10.3iT - 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
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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
−8.114921149297057689929722492808, −7.85887769354375747621864254821, −6.85721528082368581095188763116, −5.84579207439588041021327003760, −5.49812411045069714931575740782, −4.67470552623576787282584452946, −3.42063568201087327618523670994, −3.17978656405344119400592734463, −1.78123070163247051046856331641, −1.01168657861581746451372445020,
0.952772847692637523358534820204, 1.77064894237902568981103887491, 2.52067518244370899344487873131, 4.09982045233401948721037458130, 4.50374931893853714082458011941, 5.12623979800253177214779860042, 5.78960912690700619262834465151, 6.98749301207474012842580721259, 7.57028804466235217894471372739, 8.113335969706371668842017139505
