| L(s) = 1 | + 345.·2-s − 415.·3-s + 8.68e4·4-s + 2.71e5·5-s − 1.43e5·6-s + 3.51e6·7-s + 1.87e7·8-s − 1.41e7·9-s + 9.39e7·10-s − 7.42e7·11-s − 3.61e7·12-s + 1.30e8·13-s + 1.21e9·14-s − 1.12e8·15-s + 3.62e9·16-s − 1.89e9·17-s − 4.90e9·18-s − 4.37e9·19-s + 2.36e10·20-s − 1.46e9·21-s − 2.56e10·22-s − 2.01e10·23-s − 7.78e9·24-s + 4.32e10·25-s + 4.50e10·26-s + 1.18e10·27-s + 3.05e11·28-s + ⋯ |
| L(s) = 1 | + 1.91·2-s − 0.109·3-s + 2.65·4-s + 1.55·5-s − 0.209·6-s + 1.61·7-s + 3.15·8-s − 0.987·9-s + 2.97·10-s − 1.14·11-s − 0.291·12-s + 0.575·13-s + 3.07·14-s − 0.170·15-s + 3.37·16-s − 1.11·17-s − 1.88·18-s − 1.12·19-s + 4.12·20-s − 0.176·21-s − 2.19·22-s − 1.23·23-s − 0.346·24-s + 1.41·25-s + 1.09·26-s + 0.218·27-s + 4.27·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(9.135070501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.135070501\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + 1.72e10T \) |
| good | 2 | \( 1 - 345.T + 3.27e4T^{2} \) |
| 3 | \( 1 + 415.T + 1.43e7T^{2} \) |
| 5 | \( 1 - 2.71e5T + 3.05e10T^{2} \) |
| 7 | \( 1 - 3.51e6T + 4.74e12T^{2} \) |
| 11 | \( 1 + 7.42e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 1.30e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 1.89e9T + 2.86e18T^{2} \) |
| 19 | \( 1 + 4.37e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + 2.01e10T + 2.66e20T^{2} \) |
| 31 | \( 1 - 9.47e10T + 2.34e22T^{2} \) |
| 37 | \( 1 - 2.77e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 6.78e11T + 1.55e24T^{2} \) |
| 43 | \( 1 - 2.60e12T + 3.17e24T^{2} \) |
| 47 | \( 1 + 2.12e12T + 1.20e25T^{2} \) |
| 53 | \( 1 + 1.06e13T + 7.31e25T^{2} \) |
| 59 | \( 1 - 1.19e13T + 3.65e26T^{2} \) |
| 61 | \( 1 + 2.07e13T + 6.02e26T^{2} \) |
| 67 | \( 1 - 4.31e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 4.28e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 5.58e13T + 8.90e27T^{2} \) |
| 79 | \( 1 + 1.19e13T + 2.91e28T^{2} \) |
| 83 | \( 1 + 3.55e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + 6.42e14T + 1.74e29T^{2} \) |
| 97 | \( 1 - 1.38e15T + 6.33e29T^{2} \) |
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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
−13.83139717184891241557063795478, −12.87094722627055108749821300697, −11.34571096678558724468030208936, −10.62876872330980414032212232679, −8.198597981800083584535568288899, −6.27472677872991196999106793310, −5.49118823132521412281247875728, −4.52557625540745811577218258856, −2.49786217917639233662482455300, −1.86145345050667399801914061356,
1.86145345050667399801914061356, 2.49786217917639233662482455300, 4.52557625540745811577218258856, 5.49118823132521412281247875728, 6.27472677872991196999106793310, 8.198597981800083584535568288899, 10.62876872330980414032212232679, 11.34571096678558724468030208936, 12.87094722627055108749821300697, 13.83139717184891241557063795478
