| L(s) = 1 | + 1.61i·3-s + 0.618i·7-s + 0.381·9-s + 3.23·11-s + 3.23i·13-s − 2.76i·17-s + 1.23·19-s − 1.00·21-s + 3.38i·23-s + 5.47i·27-s − 2.85·29-s + 7.23·31-s + 5.23i·33-s − 6i·37-s − 5.23·39-s + ⋯ |
| L(s) = 1 | + 0.934i·3-s + 0.233i·7-s + 0.127·9-s + 0.975·11-s + 0.897i·13-s − 0.670i·17-s + 0.283·19-s − 0.218·21-s + 0.705i·23-s + 1.05i·27-s − 0.529·29-s + 1.29·31-s + 0.911i·33-s − 0.986i·37-s − 0.838·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.19585 + 1.19585i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19585 + 1.19585i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 1.61iT - 3T^{2} \) |
| 7 | \( 1 - 0.618iT - 7T^{2} \) |
| 11 | \( 1 - 3.23T + 11T^{2} \) |
| 13 | \( 1 - 3.23iT - 13T^{2} \) |
| 17 | \( 1 + 2.76iT - 17T^{2} \) |
| 19 | \( 1 - 1.23T + 19T^{2} \) |
| 23 | \( 1 - 3.38iT - 23T^{2} \) |
| 29 | \( 1 + 2.85T + 29T^{2} \) |
| 31 | \( 1 - 7.23T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 8.85T + 41T^{2} \) |
| 43 | \( 1 - 11.3iT - 43T^{2} \) |
| 47 | \( 1 - 2.09iT - 47T^{2} \) |
| 53 | \( 1 - 12.9iT - 53T^{2} \) |
| 59 | \( 1 - 8.18T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 + 1.52iT - 67T^{2} \) |
| 71 | \( 1 + 3.70T + 71T^{2} \) |
| 73 | \( 1 + 9.70iT - 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 + 2.32iT - 83T^{2} \) |
| 89 | \( 1 + 1.85T + 89T^{2} \) |
| 97 | \( 1 - 12.1iT - 97T^{2} \) |
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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
−10.00629481656044706211448846430, −9.344048964637458918160061357629, −8.922629222107300650998985715111, −7.63162431568358340979788386508, −6.79655324008673350400095147146, −5.82907649089831400098132630087, −4.72220971331063801245441562398, −4.10692224534415247062735223057, −3.05438101984641806740335652524, −1.51145070174413223116211803709,
0.869335439324012763874107286200, 1.99385160439284818133141835083, 3.37121805329453296806857762170, 4.40679409417757325569720337987, 5.62285358055517060368428942471, 6.58647972804552624038390987617, 7.09765473367091856172408955342, 8.121044136164321722545490487924, 8.678306309175585528895902649507, 9.954571452437801223826334986121
