| L(s) = 1 | − 30.5·2-s − 194.·3-s + 422.·4-s + 941.·5-s + 5.95e3·6-s − 4.51e3·7-s + 2.72e3·8-s + 1.83e4·9-s − 2.87e4·10-s + 4.15e4·11-s − 8.24e4·12-s + 1.01e5·13-s + 1.38e5·14-s − 1.83e5·15-s − 2.99e5·16-s − 5.59e5·18-s + 8.78e5·19-s + 3.97e5·20-s + 8.79e5·21-s − 1.27e6·22-s − 1.58e6·23-s − 5.31e5·24-s − 1.06e6·25-s − 3.09e6·26-s + 2.67e5·27-s − 1.90e6·28-s + 1.94e6·29-s + ⋯ |
| L(s) = 1 | − 1.35·2-s − 1.38·3-s + 0.825·4-s + 0.673·5-s + 1.87·6-s − 0.710·7-s + 0.235·8-s + 0.930·9-s − 0.909·10-s + 0.856·11-s − 1.14·12-s + 0.983·13-s + 0.960·14-s − 0.935·15-s − 1.14·16-s − 1.25·18-s + 1.54·19-s + 0.556·20-s + 0.987·21-s − 1.15·22-s − 1.17·23-s − 0.326·24-s − 0.546·25-s − 1.32·26-s + 0.0968·27-s − 0.586·28-s + 0.510·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 \) |
| good | 2 | \( 1 + 30.5T + 512T^{2} \) |
| 3 | \( 1 + 194.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 941.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 4.51e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.15e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.01e5T + 1.06e10T^{2} \) |
| 19 | \( 1 - 8.78e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.58e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.94e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.44e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 9.92e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.03e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.15e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.96e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.85e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 3.51e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.88e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.08e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 4.89e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.30e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.59e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.12e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.05e9T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.29e9T + 7.60e17T^{2} \) |
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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
−9.807516753249271206191707315987, −9.118308668668473902503178323140, −7.922984909451762265350593197477, −6.67152481871159256796025359468, −6.16699728522291953655542524948, −5.09653466857293125635154345493, −3.60412900437198496008766395492, −1.75588859832680821718344771432, −0.947030060707247762368843980657, 0,
0.947030060707247762368843980657, 1.75588859832680821718344771432, 3.60412900437198496008766395492, 5.09653466857293125635154345493, 6.16699728522291953655542524948, 6.67152481871159256796025359468, 7.922984909451762265350593197477, 9.118308668668473902503178323140, 9.807516753249271206191707315987
