| L(s) = 1 | + 6.32·3-s − 25·5-s − 44.2·7-s − 203·9-s − 720.·11-s + 146·13-s − 158.·15-s − 702·17-s + 2.73e3·19-s − 280·21-s + 4.09e3·23-s + 625·25-s − 2.82e3·27-s + 4.01e3·29-s − 4.56e3·31-s − 4.56e3·33-s + 1.10e3·35-s + 1.47e4·37-s + 923.·39-s − 4.35e3·41-s + 1.24e4·43-s + 5.07e3·45-s − 6.01e3·47-s − 1.48e4·49-s − 4.43e3·51-s + 1.81e4·53-s + 1.80e4·55-s + ⋯ |
| L(s) = 1 | + 0.405·3-s − 0.447·5-s − 0.341·7-s − 0.835·9-s − 1.79·11-s + 0.239·13-s − 0.181·15-s − 0.589·17-s + 1.73·19-s − 0.138·21-s + 1.61·23-s + 0.200·25-s − 0.744·27-s + 0.885·29-s − 0.853·31-s − 0.728·33-s + 0.152·35-s + 1.77·37-s + 0.0972·39-s − 0.404·41-s + 1.02·43-s + 0.373·45-s − 0.397·47-s − 0.883·49-s − 0.239·51-s + 0.887·53-s + 0.803·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.509982416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.509982416\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| good | 3 | \( 1 - 6.32T + 243T^{2} \) |
| 7 | \( 1 + 44.2T + 1.68e4T^{2} \) |
| 11 | \( 1 + 720.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 146T + 3.71e5T^{2} \) |
| 17 | \( 1 + 702T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.73e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.09e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.01e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.56e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.47e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.35e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.24e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.01e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.81e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.97e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.21e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.61e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.54e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.62e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.67e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.87e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.05e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.68e4T + 8.58e9T^{2} \) |
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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
−10.93311781962910836716675180448, −9.802471878091752943783771775280, −8.839626604553640661833342998693, −7.955254604777747824361998736997, −7.17746656229355761424452603713, −5.74475709357530976347433880492, −4.85878873764652386062272315053, −3.26337482456587039544030833630, −2.61585971034506651945096348300, −0.64567714477605399038840036874,
0.64567714477605399038840036874, 2.61585971034506651945096348300, 3.26337482456587039544030833630, 4.85878873764652386062272315053, 5.74475709357530976347433880492, 7.17746656229355761424452603713, 7.955254604777747824361998736997, 8.839626604553640661833342998693, 9.802471878091752943783771775280, 10.93311781962910836716675180448
