| L(s) = 1 | + 2-s + 4-s − 1.37·5-s + 8-s − 1.37·10-s − 4.37·11-s − 2·13-s + 16-s − 4.37·17-s − 5·19-s − 1.37·20-s − 4.37·22-s + 7.37·23-s − 3.11·25-s − 2·26-s − 2.74·29-s − 2·31-s + 32-s − 4.37·34-s + 2·37-s − 5·38-s − 1.37·40-s + 10.3·41-s + 9.11·43-s − 4.37·44-s + 7.37·46-s − 3.11·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.613·5-s + 0.353·8-s − 0.433·10-s − 1.31·11-s − 0.554·13-s + 0.250·16-s − 1.06·17-s − 1.14·19-s − 0.306·20-s − 0.932·22-s + 1.53·23-s − 0.623·25-s − 0.392·26-s − 0.509·29-s − 0.359·31-s + 0.176·32-s − 0.749·34-s + 0.328·37-s − 0.811·38-s − 0.216·40-s + 1.61·41-s + 1.39·43-s − 0.659·44-s + 1.08·46-s − 0.440·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.921915645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.921915645\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.37T + 5T^{2} \) |
| 11 | \( 1 + 4.37T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 4.37T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 - 7.37T + 23T^{2} \) |
| 29 | \( 1 + 2.74T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 9.11T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 2.74T + 53T^{2} \) |
| 59 | \( 1 - 7.11T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 5.11T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 + 5.48T + 83T^{2} \) |
| 89 | \( 1 - 3.25T + 89T^{2} \) |
| 97 | \( 1 + 9.11T + 97T^{2} \) |
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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
−7.65398674743614503654711371186, −7.20301508429689924892448788879, −6.44269782175383054909055964766, −5.61707352755937392141007599464, −4.97525554524212757707466908115, −4.31794644608664716469302008939, −3.67007879400024221145789832056, −2.56799111915773002711713205595, −2.23348000867179067521673549461, −0.58079529470835522575617371732,
0.58079529470835522575617371732, 2.23348000867179067521673549461, 2.56799111915773002711713205595, 3.67007879400024221145789832056, 4.31794644608664716469302008939, 4.97525554524212757707466908115, 5.61707352755937392141007599464, 6.44269782175383054909055964766, 7.20301508429689924892448788879, 7.65398674743614503654711371186
