| L(s) = 1 | − 3.06i·5-s + 1.53·7-s + 4.08i·11-s − i·13-s − 4.47·17-s + 2.68i·19-s − 3.24·23-s − 4.39·25-s − 8.41i·29-s + 5.14·31-s − 4.71i·35-s − 5.60i·37-s + 5.34·41-s − 8.91i·43-s − 10.6·47-s + ⋯ |
| L(s) = 1 | − 1.37i·5-s + 0.581·7-s + 1.23i·11-s − 0.277i·13-s − 1.08·17-s + 0.615i·19-s − 0.676·23-s − 0.878·25-s − 1.56i·29-s + 0.924·31-s − 0.797i·35-s − 0.921i·37-s + 0.834·41-s − 1.35i·43-s − 1.54·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.150347462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.150347462\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 5 | \( 1 + 3.06iT - 5T^{2} \) |
| 7 | \( 1 - 1.53T + 7T^{2} \) |
| 11 | \( 1 - 4.08iT - 11T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 - 2.68iT - 19T^{2} \) |
| 23 | \( 1 + 3.24T + 23T^{2} \) |
| 29 | \( 1 + 8.41iT - 29T^{2} \) |
| 31 | \( 1 - 5.14T + 31T^{2} \) |
| 37 | \( 1 + 5.60iT - 37T^{2} \) |
| 41 | \( 1 - 5.34T + 41T^{2} \) |
| 43 | \( 1 + 8.91iT - 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 2.48iT - 53T^{2} \) |
| 59 | \( 1 + 3.09iT - 59T^{2} \) |
| 61 | \( 1 + 8.47iT - 61T^{2} \) |
| 67 | \( 1 + 9.32iT - 67T^{2} \) |
| 71 | \( 1 + 7.52T + 71T^{2} \) |
| 73 | \( 1 + 5.60T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 4.46iT - 83T^{2} \) |
| 89 | \( 1 + 5.90T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−8.059787113780820607902956275423, −7.85559531047656295845839495607, −6.72291722202656633293875274089, −5.91256132818113923165617456508, −4.97633726406801918411397526290, −4.55622241237596130743152014767, −3.84727722686616670382697435377, −2.27582953554115907560369318999, −1.65124187011667179333836275057, −0.32838887142642900224533563977,
1.37570950629206497724266990606, 2.64213224000829654663123685424, 3.11923206933633767867149966862, 4.18572465219504674507050892946, 4.99574827928459473425352516664, 6.07855705507035449880619408646, 6.55408258076256985940007440544, 7.21760983398875895470936675148, 8.124147488424957566726518210353, 8.654616647422071098588328123125
