| L(s) = 1 | + 3-s − 5-s − 2·9-s + 4·11-s − 13-s − 15-s + 3·17-s − 5·23-s − 4·25-s − 5·27-s + 7·29-s − 4·31-s + 4·33-s + 10·37-s − 39-s − 5·41-s + 5·43-s + 2·45-s + 7·47-s − 7·49-s + 3·51-s + 11·53-s − 4·55-s − 3·59-s + 11·61-s + 65-s + 3·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 2/3·9-s + 1.20·11-s − 0.277·13-s − 0.258·15-s + 0.727·17-s − 1.04·23-s − 4/5·25-s − 0.962·27-s + 1.29·29-s − 0.718·31-s + 0.696·33-s + 1.64·37-s − 0.160·39-s − 0.780·41-s + 0.762·43-s + 0.298·45-s + 1.02·47-s − 49-s + 0.420·51-s + 1.51·53-s − 0.539·55-s − 0.390·59-s + 1.40·61-s + 0.124·65-s + 0.366·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.101210392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.101210392\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 11 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−8.070223875508849209271112014483, −7.63607991647119737095362961926, −6.69614388410678726347088668218, −6.02249493803915114497163101839, −5.28968462197699658502621172591, −4.14806492488521143828856305471, −3.76507390629787346769837834466, −2.83397001973401195812410569621, −1.98100308025709555986426797089, −0.74353520954790544832755635907,
0.74353520954790544832755635907, 1.98100308025709555986426797089, 2.83397001973401195812410569621, 3.76507390629787346769837834466, 4.14806492488521143828856305471, 5.28968462197699658502621172591, 6.02249493803915114497163101839, 6.69614388410678726347088668218, 7.63607991647119737095362961926, 8.070223875508849209271112014483
