| L(s) = 1 | + 44.6·2-s − 51.0·3-s + 1.47e3·4-s − 151.·5-s − 2.27e3·6-s − 5.43e3·7-s + 4.31e4·8-s − 1.70e4·9-s − 6.74e3·10-s − 3.38e4·11-s − 7.54e4·12-s − 2.42e5·14-s + 7.71e3·15-s + 1.16e6·16-s − 5.88e5·17-s − 7.62e5·18-s + 1.23e5·19-s − 2.23e5·20-s + 2.77e5·21-s − 1.51e6·22-s − 1.55e6·23-s − 2.20e6·24-s − 1.93e6·25-s + 1.87e6·27-s − 8.04e6·28-s + 2.73e5·29-s + 3.44e5·30-s + ⋯ |
| L(s) = 1 | + 1.97·2-s − 0.363·3-s + 2.88·4-s − 0.108·5-s − 0.717·6-s − 0.855·7-s + 3.72·8-s − 0.867·9-s − 0.213·10-s − 0.696·11-s − 1.05·12-s − 1.68·14-s + 0.0393·15-s + 4.45·16-s − 1.71·17-s − 1.71·18-s + 0.216·19-s − 0.312·20-s + 0.311·21-s − 1.37·22-s − 1.15·23-s − 1.35·24-s − 0.988·25-s + 0.679·27-s − 2.47·28-s + 0.0719·29-s + 0.0775·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 - 44.6T + 512T^{2} \) |
| 3 | \( 1 + 51.0T + 1.96e4T^{2} \) |
| 5 | \( 1 + 151.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 5.43e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.38e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 5.88e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.23e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.55e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.73e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.07e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 8.27e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.69e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.23e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.71e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.85e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 2.83e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 2.89e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 9.52e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.35e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.20e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.80e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.16e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.00e9T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.99e8T + 7.60e17T^{2} \) |
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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
−11.09140919123360700880358878858, −10.08658990464225919598801540343, −8.177444379757210569547643378660, −6.81771021583112483600610350337, −6.12422590326429875583262375866, −5.21266029606430997500689457507, −4.12156070933077662387073169292, −3.02848665352434925060751580562, −2.11235660075551636512891955672, 0,
2.11235660075551636512891955672, 3.02848665352434925060751580562, 4.12156070933077662387073169292, 5.21266029606430997500689457507, 6.12422590326429875583262375866, 6.81771021583112483600610350337, 8.177444379757210569547643378660, 10.08658990464225919598801540343, 11.09140919123360700880358878858
