| L(s) = 1 | + 128·2-s − 2.18e3·3-s + 1.63e4·4-s − 8.04e4·5-s − 2.79e5·6-s + 4.25e5·7-s + 2.09e6·8-s + 4.78e6·9-s − 1.03e7·10-s + 1.87e7·11-s − 3.58e7·12-s − 6.27e7·13-s + 5.44e7·14-s + 1.76e8·15-s + 2.68e8·16-s − 1.90e9·17-s + 6.12e8·18-s + 4.03e8·19-s − 1.31e9·20-s − 9.30e8·21-s + 2.40e9·22-s + 2.06e10·23-s − 4.58e9·24-s − 2.40e10·25-s − 8.03e9·26-s − 1.04e10·27-s + 6.97e9·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.460·5-s − 0.408·6-s + 0.195·7-s + 0.353·8-s + 0.333·9-s − 0.325·10-s + 0.290·11-s − 0.288·12-s − 0.277·13-s + 0.138·14-s + 0.265·15-s + 0.250·16-s − 1.12·17-s + 0.235·18-s + 0.103·19-s − 0.230·20-s − 0.112·21-s + 0.205·22-s + 1.26·23-s − 0.204·24-s − 0.787·25-s − 0.196·26-s − 0.192·27-s + 0.0976·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 128T \) |
| 3 | \( 1 + 2.18e3T \) |
| 13 | \( 1 + 6.27e7T \) |
| good | 5 | \( 1 + 8.04e4T + 3.05e10T^{2} \) |
| 7 | \( 1 - 4.25e5T + 4.74e12T^{2} \) |
| 11 | \( 1 - 1.87e7T + 4.17e15T^{2} \) |
| 17 | \( 1 + 1.90e9T + 2.86e18T^{2} \) |
| 19 | \( 1 - 4.03e8T + 1.51e19T^{2} \) |
| 23 | \( 1 - 2.06e10T + 2.66e20T^{2} \) |
| 29 | \( 1 - 5.56e10T + 8.62e21T^{2} \) |
| 31 | \( 1 - 2.47e11T + 2.34e22T^{2} \) |
| 37 | \( 1 + 3.26e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 1.74e12T + 1.55e24T^{2} \) |
| 43 | \( 1 - 2.34e12T + 3.17e24T^{2} \) |
| 47 | \( 1 + 4.04e12T + 1.20e25T^{2} \) |
| 53 | \( 1 - 1.37e13T + 7.31e25T^{2} \) |
| 59 | \( 1 + 3.24e12T + 3.65e26T^{2} \) |
| 61 | \( 1 + 4.49e13T + 6.02e26T^{2} \) |
| 67 | \( 1 - 8.68e13T + 2.46e27T^{2} \) |
| 71 | \( 1 + 4.94e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + 1.72e14T + 8.90e27T^{2} \) |
| 79 | \( 1 + 2.64e14T + 2.91e28T^{2} \) |
| 83 | \( 1 + 2.90e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + 3.43e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 7.85e14T + 6.33e29T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.26732991080079620806155116602, −10.10554740737834991466799449521, −8.594895273896312142726458835148, −7.23669316420554249743064351497, −6.32455416657873170619973215839, −5.01057427319683109582010884308, −4.19029324698499320408484023098, −2.81368408913742127548972492668, −1.37410863845763963938254475682, 0,
1.37410863845763963938254475682, 2.81368408913742127548972492668, 4.19029324698499320408484023098, 5.01057427319683109582010884308, 6.32455416657873170619973215839, 7.23669316420554249743064351497, 8.594895273896312142726458835148, 10.10554740737834991466799449521, 11.26732991080079620806155116602
