| L(s) = 1 | + 27·3-s − 282.·5-s − 1.36e3·7-s + 729·9-s − 3.81e3·11-s − 1.48e4·13-s − 7.62e3·15-s − 4.63e3·17-s + 3.21e3·19-s − 3.67e4·21-s − 3.93e4·23-s + 1.54e3·25-s + 1.96e4·27-s − 1.54e5·29-s − 7.12e4·31-s − 1.02e5·33-s + 3.84e5·35-s + 2.61e5·37-s − 4.01e5·39-s + 3.25e5·41-s − 5.31e5·43-s − 2.05e5·45-s − 8.95e5·47-s + 1.03e6·49-s − 1.25e5·51-s − 1.02e6·53-s + 1.07e6·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.00·5-s − 1.50·7-s + 0.333·9-s − 0.864·11-s − 1.87·13-s − 0.583·15-s − 0.228·17-s + 0.107·19-s − 0.866·21-s − 0.674·23-s + 0.0198·25-s + 0.192·27-s − 1.17·29-s − 0.429·31-s − 0.498·33-s + 1.51·35-s + 0.850·37-s − 1.08·39-s + 0.737·41-s − 1.02·43-s − 0.336·45-s − 1.25·47-s + 1.25·49-s − 0.132·51-s − 0.941·53-s + 0.872·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.1943550572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1943550572\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 27T \) |
| good | 5 | \( 1 + 282.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.36e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.81e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.48e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 4.63e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.21e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.93e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.54e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 7.12e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.61e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.25e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.31e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.95e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.02e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.89e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.00e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.51e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.15e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.35e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.62e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.94e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.21e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.04e7T + 8.07e13T^{2} \) |
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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
−9.775786369452353240328337504559, −9.550877577275062461577505823315, −8.088495932558763969602992435000, −7.50718541830680290794521264210, −6.61293255698707023764573730062, −5.23293103897029616255171123883, −4.05149283540014872562027599242, −3.13911126168384688155837578521, −2.26133327035553586210002933014, −0.18223409554208851928775608897,
0.18223409554208851928775608897, 2.26133327035553586210002933014, 3.13911126168384688155837578521, 4.05149283540014872562027599242, 5.23293103897029616255171123883, 6.61293255698707023764573730062, 7.50718541830680290794521264210, 8.088495932558763969602992435000, 9.550877577275062461577505823315, 9.775786369452353240328337504559
