| L(s) = 1 | + 27·3-s + 368.·5-s + 1.05e3·7-s + 729·9-s + 4.72e3·11-s + 5.52e3·13-s + 9.94e3·15-s + 2.84e4·17-s + 1.26e4·19-s + 2.85e4·21-s − 1.69e4·23-s + 5.76e4·25-s + 1.96e4·27-s − 7.56e4·29-s + 6.93e4·31-s + 1.27e5·33-s + 3.90e5·35-s − 2.92e4·37-s + 1.49e5·39-s + 2.95e5·41-s + 1.63e4·43-s + 2.68e5·45-s + 6.81e4·47-s + 2.97e5·49-s + 7.69e5·51-s − 1.85e6·53-s + 1.74e6·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.31·5-s + 1.16·7-s + 0.333·9-s + 1.07·11-s + 0.697·13-s + 0.761·15-s + 1.40·17-s + 0.422·19-s + 0.673·21-s − 0.291·23-s + 0.737·25-s + 0.192·27-s − 0.576·29-s + 0.418·31-s + 0.618·33-s + 1.53·35-s − 0.0948·37-s + 0.402·39-s + 0.670·41-s + 0.0314·43-s + 0.439·45-s + 0.0957·47-s + 0.361·49-s + 0.811·51-s − 1.71·53-s + 1.41·55-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
| \(L(4)\) |
\(\approx\) |
\(5.424054632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.424054632\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 368.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.05e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.72e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.52e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.84e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.26e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.69e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 7.56e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 6.93e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.92e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.95e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.63e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 6.81e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.85e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.75e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.77e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.76e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.66e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 6.38e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.32e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.05e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.79e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.84e6T + 8.07e13T^{2} \) |
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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
−9.939656778362885539898284987252, −9.309589023031062427314154856073, −8.385315429569370886409263634222, −7.49302499642291105421949325044, −6.22946614992029472051936611038, −5.44971372224529095307817240858, −4.25457972044535799854869706055, −3.03786228667935104125396053449, −1.66334816451859577170085496987, −1.28689340606081124153514402503,
1.28689340606081124153514402503, 1.66334816451859577170085496987, 3.03786228667935104125396053449, 4.25457972044535799854869706055, 5.44971372224529095307817240858, 6.22946614992029472051936611038, 7.49302499642291105421949325044, 8.385315429569370886409263634222, 9.309589023031062427314154856073, 9.939656778362885539898284987252
