| L(s) = 1 | + 2i·3-s − 6i·7-s + 23·9-s − 32·11-s + 38i·13-s + 26i·17-s + 100·19-s + 12·21-s − 78i·23-s + 100i·27-s + 50·29-s + 108·31-s − 64i·33-s + 266i·37-s − 76·39-s + ⋯ |
| L(s) = 1 | + 0.384i·3-s − 0.323i·7-s + 0.851·9-s − 0.877·11-s + 0.810i·13-s + 0.370i·17-s + 1.20·19-s + 0.124·21-s − 0.707i·23-s + 0.712i·27-s + 0.320·29-s + 0.625·31-s − 0.337i·33-s + 1.18i·37-s − 0.312·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.841899855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.841899855\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2iT - 27T^{2} \) |
| 7 | \( 1 + 6iT - 343T^{2} \) |
| 11 | \( 1 + 32T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 26iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 100T + 6.85e3T^{2} \) |
| 23 | \( 1 + 78iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 50T + 2.43e4T^{2} \) |
| 31 | \( 1 - 108T + 2.97e4T^{2} \) |
| 37 | \( 1 - 266iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 22T + 6.89e4T^{2} \) |
| 43 | \( 1 - 442iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 514iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 2iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 500T + 2.05e5T^{2} \) |
| 61 | \( 1 + 518T + 2.26e5T^{2} \) |
| 67 | \( 1 + 126iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 412T + 3.57e5T^{2} \) |
| 73 | \( 1 - 878iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 600T + 4.93e5T^{2} \) |
| 83 | \( 1 - 282iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 150T + 7.04e5T^{2} \) |
| 97 | \( 1 - 386iT - 9.12e5T^{2} \) |
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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
−10.86475004937360218547711231027, −10.08554000731444361154593768878, −9.403824827627716724822132422354, −8.188848125144498443810314292992, −7.31379971720802691878323149035, −6.31702851948624255243191570083, −4.97318672059577961980651840765, −4.19590710006785558926753365408, −2.84817102981007986351160784742, −1.23295451945082391674409910234,
0.71160351408492586945971911327, 2.23618783113428555914189139512, 3.50158159467581486014116015405, 4.98085989018928893461698109074, 5.79271527161043583096136312479, 7.17941231620303986546456277637, 7.69416537364798965174973992097, 8.824612195946698976107674073743, 9.917023165950861455861547693748, 10.54673404148715868669198788016
