| L(s) = 1 | − 22.2·3-s − 24.1·5-s − 222.·7-s + 254.·9-s − 158.·11-s − 403.·13-s + 537.·15-s − 1.69e3·17-s + 2.41e3·19-s + 4.95e3·21-s − 3.39e3·23-s − 2.54e3·25-s − 245.·27-s + 3.24e3·29-s + 2.20e3·31-s + 3.53e3·33-s + 5.35e3·35-s + 2.63e3·37-s + 8.99e3·39-s − 4.26e3·41-s − 6.73e3·43-s − 6.12e3·45-s + 2.89e3·47-s + 3.25e4·49-s + 3.78e4·51-s + 3.00e4·53-s + 3.82e3·55-s + ⋯ |
| L(s) = 1 | − 1.43·3-s − 0.431·5-s − 1.71·7-s + 1.04·9-s − 0.395·11-s − 0.662·13-s + 0.617·15-s − 1.42·17-s + 1.53·19-s + 2.45·21-s − 1.33·23-s − 0.813·25-s − 0.0648·27-s + 0.715·29-s + 0.411·31-s + 0.565·33-s + 0.739·35-s + 0.316·37-s + 0.947·39-s − 0.395·41-s − 0.555·43-s − 0.451·45-s + 0.191·47-s + 1.93·49-s + 2.03·51-s + 1.47·53-s + 0.170·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 257 | \( 1 - 6.60e4T \) |
| good | 3 | \( 1 + 22.2T + 243T^{2} \) |
| 5 | \( 1 + 24.1T + 3.12e3T^{2} \) |
| 7 | \( 1 + 222.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 158.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 403.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.69e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.41e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.39e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.24e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.20e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.63e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.26e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.73e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.89e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.00e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.97e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.15e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.14e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.51e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.35e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.33e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.27e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.93e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.01e5T + 8.58e9T^{2} \) |
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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
−8.966601039276983291191415077766, −7.70969498596634935907954161250, −6.85492527871158065245435886444, −6.25800931439072045888206883913, −5.49284988353827720310580792001, −4.53306325897123683090419056832, −3.52254098359476097061512948814, −2.40065520541199370323996554287, −0.64818581547178742158833969982, 0,
0.64818581547178742158833969982, 2.40065520541199370323996554287, 3.52254098359476097061512948814, 4.53306325897123683090419056832, 5.49284988353827720310580792001, 6.25800931439072045888206883913, 6.85492527871158065245435886444, 7.70969498596634935907954161250, 8.966601039276983291191415077766
