| L(s) = 1 | + 27·3-s − 40.1·5-s + 729·9-s + 4.18e3·11-s − 508.·13-s − 1.08e3·15-s − 1.06e4·17-s + 1.79e4·19-s − 6.66e4·23-s − 7.65e4·25-s + 1.96e4·27-s + 3.44e4·29-s − 4.61e4·31-s + 1.13e5·33-s − 4.23e5·37-s − 1.37e4·39-s − 5.00e5·41-s + 5.69e5·43-s − 2.92e4·45-s + 5.07e5·47-s − 2.86e5·51-s + 9.87e5·53-s − 1.68e5·55-s + 4.85e5·57-s − 5.79e5·59-s + 1.89e5·61-s + 2.04e4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.143·5-s + 0.333·9-s + 0.948·11-s − 0.0642·13-s − 0.0829·15-s − 0.523·17-s + 0.601·19-s − 1.14·23-s − 0.979·25-s + 0.192·27-s + 0.262·29-s − 0.278·31-s + 0.547·33-s − 1.37·37-s − 0.0370·39-s − 1.13·41-s + 1.09·43-s − 0.0478·45-s + 0.712·47-s − 0.302·51-s + 0.911·53-s − 0.136·55-s + 0.347·57-s − 0.367·59-s + 0.106·61-s + 0.00922·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 27T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 40.1T + 7.81e4T^{2} \) |
| 11 | \( 1 - 4.18e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 508.T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.06e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.79e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.66e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 3.44e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 4.61e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.23e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.00e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.69e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.07e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 9.87e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 5.79e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.89e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.75e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.43e3T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.72e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.68e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.83e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.28e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.50e6T + 8.07e13T^{2} \) |
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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
−9.117029591447356533348194292012, −8.315190607592594022418410963578, −7.40502274497656227281806897604, −6.55387798117867356003454257407, −5.49238297763557497701300039195, −4.22891303861983973417539323576, −3.56316515435135889150165610359, −2.31102001834525998471815361512, −1.35395498769185484508126362680, 0,
1.35395498769185484508126362680, 2.31102001834525998471815361512, 3.56316515435135889150165610359, 4.22891303861983973417539323576, 5.49238297763557497701300039195, 6.55387798117867356003454257407, 7.40502274497656227281806897604, 8.315190607592594022418410963578, 9.117029591447356533348194292012
