| L(s) = 1 | − 8·2-s + 14.3·3-s + 64·4-s + 230.·5-s − 115.·6-s − 1.45e3·7-s − 512·8-s − 1.98e3·9-s − 1.84e3·10-s − 6.48e3·11-s + 920.·12-s − 1.14e3·13-s + 1.16e4·14-s + 3.31e3·15-s + 4.09e3·16-s − 1.01e4·17-s + 1.58e4·18-s − 3.16e4·19-s + 1.47e4·20-s − 2.09e4·21-s + 5.18e4·22-s − 2.13e4·23-s − 7.36e3·24-s − 2.48e4·25-s + 9.13e3·26-s − 5.99e4·27-s − 9.33e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.307·3-s + 0.5·4-s + 0.825·5-s − 0.217·6-s − 1.60·7-s − 0.353·8-s − 0.905·9-s − 0.583·10-s − 1.46·11-s + 0.153·12-s − 0.144·13-s + 1.13·14-s + 0.253·15-s + 0.250·16-s − 0.498·17-s + 0.640·18-s − 1.05·19-s + 0.412·20-s − 0.494·21-s + 1.03·22-s − 0.365·23-s − 0.108·24-s − 0.318·25-s + 0.101·26-s − 0.585·27-s − 0.803·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.04415755900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04415755900\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 269 | \( 1 + 1.94e7T \) |
| good | 3 | \( 1 - 14.3T + 2.18e3T^{2} \) |
| 5 | \( 1 - 230.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.45e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 6.48e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.14e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.01e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.16e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.13e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.89e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.55e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.02e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.20e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.31e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.21e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.77e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 7.34e4T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.56e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.28e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.47e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.65e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.53e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.34e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.44e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 2.72e5T + 8.07e13T^{2} \) |
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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
−9.532346055413128932297722255061, −9.038444926715759899525593869464, −8.038476086671851981400021966097, −7.04255712267325465498999655445, −6.04415644493371230907431145810, −5.49251350833277353089093645947, −3.67783107496985387406487030819, −2.63505983142150258523696291238, −2.06572409139643410995172418641, −0.089845587069360355831667030783,
0.089845587069360355831667030783, 2.06572409139643410995172418641, 2.63505983142150258523696291238, 3.67783107496985387406487030819, 5.49251350833277353089093645947, 6.04415644493371230907431145810, 7.04255712267325465498999655445, 8.038476086671851981400021966097, 9.038444926715759899525593869464, 9.532346055413128932297722255061
