| L(s) = 1 | + 13.5·3-s + 94.9·5-s − 59·9-s − 476·11-s − 963.·13-s + 1.28e3·15-s + 895.·17-s + 637.·19-s − 3.69e3·23-s + 5.89e3·25-s − 4.09e3·27-s + 1.39e3·29-s + 1.92e3·31-s − 6.45e3·33-s + 1.20e4·37-s − 1.30e4·39-s − 1.52e4·41-s − 9.72e3·43-s − 5.60e3·45-s − 2.92e4·47-s + 1.21e4·51-s + 4.31e3·53-s − 4.51e4·55-s + 8.64e3·57-s − 2.08e4·59-s − 9.29e3·61-s − 9.14e4·65-s + ⋯ |
| L(s) = 1 | + 0.870·3-s + 1.69·5-s − 0.242·9-s − 1.18·11-s − 1.58·13-s + 1.47·15-s + 0.751·17-s + 0.405·19-s − 1.45·23-s + 1.88·25-s − 1.08·27-s + 0.307·29-s + 0.359·31-s − 1.03·33-s + 1.45·37-s − 1.37·39-s − 1.41·41-s − 0.801·43-s − 0.412·45-s − 1.93·47-s + 0.653·51-s + 0.210·53-s − 2.01·55-s + 0.352·57-s − 0.779·59-s − 0.319·61-s − 2.68·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 13.5T + 243T^{2} \) |
| 5 | \( 1 - 94.9T + 3.12e3T^{2} \) |
| 11 | \( 1 + 476T + 1.61e5T^{2} \) |
| 13 | \( 1 + 963.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 895.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 637.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.69e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.39e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.92e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.20e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.52e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.72e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.92e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.31e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.08e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 9.29e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.02e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.97e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.12e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.31e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.54e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.07e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.77e4T + 8.58e9T^{2} \) |
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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
−9.322669104387555545816982564285, −8.209075378714748043270546395211, −7.63079710717712931775191717151, −6.37844174457453344865548462060, −5.51378274470228493187056295521, −4.83512883718446235830102126004, −3.10890741707897226267911461886, −2.48542976310443483781654637334, −1.70826239686442198587932475132, 0,
1.70826239686442198587932475132, 2.48542976310443483781654637334, 3.10890741707897226267911461886, 4.83512883718446235830102126004, 5.51378274470228493187056295521, 6.37844174457453344865548462060, 7.63079710717712931775191717151, 8.209075378714748043270546395211, 9.322669104387555545816982564285
