| L(s) = 1 | − 20.0·3-s + 33.3·5-s + 49·7-s + 159.·9-s + 660.·11-s + 145.·13-s − 668.·15-s + 435.·17-s + 1.52e3·19-s − 982.·21-s + 206.·23-s − 2.01e3·25-s + 1.68e3·27-s + 2.03e3·29-s − 1.15e3·31-s − 1.32e4·33-s + 1.63e3·35-s − 6.75e3·37-s − 2.92e3·39-s − 1.53e4·41-s + 6.25e3·43-s + 5.29e3·45-s + 1.26e4·47-s + 2.40e3·49-s − 8.72e3·51-s − 1.08e4·53-s + 2.20e4·55-s + ⋯ |
| L(s) = 1 | − 1.28·3-s + 0.596·5-s + 0.377·7-s + 0.654·9-s + 1.64·11-s + 0.239·13-s − 0.766·15-s + 0.365·17-s + 0.970·19-s − 0.486·21-s + 0.0814·23-s − 0.644·25-s + 0.444·27-s + 0.450·29-s − 0.215·31-s − 2.11·33-s + 0.225·35-s − 0.811·37-s − 0.307·39-s − 1.42·41-s + 0.516·43-s + 0.390·45-s + 0.832·47-s + 0.142·49-s − 0.469·51-s − 0.530·53-s + 0.980·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.807134511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.807134511\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 + 20.0T + 243T^{2} \) |
| 5 | \( 1 - 33.3T + 3.12e3T^{2} \) |
| 11 | \( 1 - 660.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 145.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 435.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.52e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 206.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.03e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.75e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.53e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.25e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.26e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.08e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.25e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.67e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.38e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.46e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.08e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.26e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.88e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.98e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.73e4T + 8.58e9T^{2} \) |
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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
−10.37290942857218780562680484965, −9.551301651583423306606272758956, −8.615080417251311953318602777835, −7.24402008520926145041670943667, −6.35613435471612349741653809956, −5.67009353926424589884720863296, −4.75548740622755235940270115080, −3.51867339561346735721580831021, −1.70856088688335314768078405180, −0.794798929894067617019137886423,
0.794798929894067617019137886423, 1.70856088688335314768078405180, 3.51867339561346735721580831021, 4.75548740622755235940270115080, 5.67009353926424589884720863296, 6.35613435471612349741653809956, 7.24402008520926145041670943667, 8.615080417251311953318602777835, 9.551301651583423306606272758956, 10.37290942857218780562680484965
