L(s) = 1 | − 3·3-s − 19.8·5-s + 19.8·7-s + 9·9-s + 48·11-s − 79.5·13-s + 59.6·15-s − 42·17-s − 92·19-s − 59.6·21-s − 39.7·23-s + 271·25-s − 27·27-s + 19.8·29-s − 139.·31-s − 144·33-s − 396·35-s − 198.·37-s + 238.·39-s + 6·41-s − 92·43-s − 179.·45-s + 39.7·47-s + 53·49-s + 126·51-s + 497.·53-s − 955.·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.77·5-s + 1.07·7-s + 0.333·9-s + 1.31·11-s − 1.69·13-s + 1.02·15-s − 0.599·17-s − 1.11·19-s − 0.620·21-s − 0.360·23-s + 2.16·25-s − 0.192·27-s + 0.127·29-s − 0.807·31-s − 0.759·33-s − 1.91·35-s − 0.884·37-s + 0.980·39-s + 0.0228·41-s − 0.326·43-s − 0.593·45-s + 0.123·47-s + 0.154·49-s + 0.345·51-s + 1.28·53-s − 2.34·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8532640670\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8532640670\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
good | 5 | \( 1 + 19.8T + 125T^{2} \) |
| 7 | \( 1 - 19.8T + 343T^{2} \) |
| 11 | \( 1 - 48T + 1.33e3T^{2} \) |
| 13 | \( 1 + 79.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 42T + 4.91e3T^{2} \) |
| 19 | \( 1 + 92T + 6.85e3T^{2} \) |
| 23 | \( 1 + 39.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 19.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 139.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 198.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 92T + 7.95e4T^{2} \) |
| 47 | \( 1 - 39.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 497.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 516T + 2.05e5T^{2} \) |
| 61 | \( 1 - 358.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 524T + 3.00e5T^{2} \) |
| 71 | \( 1 - 994.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 430T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 432T + 5.71e5T^{2} \) |
| 89 | \( 1 + 630T + 7.04e5T^{2} \) |
| 97 | \( 1 - 862T + 9.12e5T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.08509825993297760030522995205, −8.856260425524114584414103424322, −8.179849312979938799124331207734, −7.25219177704208574854297326674, −6.74192302299485496203504444575, −5.18186339036491029665882696164, −4.41826592676586169956660837818, −3.79964579365562355492345370841, −2.06240059032154980978798587856, −0.52789744647342078562359004105,
0.52789744647342078562359004105, 2.06240059032154980978798587856, 3.79964579365562355492345370841, 4.41826592676586169956660837818, 5.18186339036491029665882696164, 6.74192302299485496203504444575, 7.25219177704208574854297326674, 8.179849312979938799124331207734, 8.856260425524114584414103424322, 10.08509825993297760030522995205