| L(s) = 1 | − 8·2-s + 64·4-s + 427.·5-s + 345.·7-s − 512·8-s − 3.42e3·10-s − 2.18e3·11-s − 2.61e3·13-s − 2.76e3·14-s + 4.09e3·16-s − 3.79e4·17-s − 2.63e4·19-s + 2.73e4·20-s + 1.74e4·22-s + 1.21e4·23-s + 1.04e5·25-s + 2.08e4·26-s + 2.21e4·28-s − 2.50e5·29-s + 1.79e5·31-s − 3.27e4·32-s + 3.03e5·34-s + 1.47e5·35-s + 4.34e5·37-s + 2.10e5·38-s − 2.19e5·40-s + 2.00e5·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.53·5-s + 0.380·7-s − 0.353·8-s − 1.08·10-s − 0.494·11-s − 0.329·13-s − 0.269·14-s + 0.250·16-s − 1.87·17-s − 0.881·19-s + 0.765·20-s + 0.349·22-s + 0.208·23-s + 1.34·25-s + 0.233·26-s + 0.190·28-s − 1.90·29-s + 1.08·31-s − 0.176·32-s + 1.32·34-s + 0.582·35-s + 1.40·37-s + 0.623·38-s − 0.541·40-s + 0.454·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.928209705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.928209705\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 3 | \( 1 \) |
| 23 | \( 1 - 1.21e4T \) |
| good | 5 | \( 1 - 427.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 345.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.18e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 2.61e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.79e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.63e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 2.50e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.79e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.34e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.00e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 8.41e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.00e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.62e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.18e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.54e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.66e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.02e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.58e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.22e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.16e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.19e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.97e6T + 8.07e13T^{2} \) |
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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
−9.933927226645463900891329052351, −9.204815533991874014996271387225, −8.457125649837891263169361494251, −7.26179483356045328126080984780, −6.32851876371167061550312960743, −5.52191897663642095017966244691, −4.33620282863980882235194909543, −2.38807997734796050212646817697, −2.10967083682152986686508784899, −0.68610430393071211793351576646,
0.68610430393071211793351576646, 2.10967083682152986686508784899, 2.38807997734796050212646817697, 4.33620282863980882235194909543, 5.52191897663642095017966244691, 6.32851876371167061550312960743, 7.26179483356045328126080984780, 8.457125649837891263169361494251, 9.204815533991874014996271387225, 9.933927226645463900891329052351
