| L(s) = 1 | + 5.48·2-s − 27·3-s − 97.8·4-s − 94.0·5-s − 148.·6-s − 955.·7-s − 1.23e3·8-s + 729·9-s − 515.·10-s + 1.69e3·11-s + 2.64e3·12-s + 9.55e3·13-s − 5.24e3·14-s + 2.53e3·15-s + 5.73e3·16-s − 2.75e4·17-s + 3.99e3·18-s − 4.35e4·19-s + 9.20e3·20-s + 2.58e4·21-s + 9.29e3·22-s − 2.31e4·23-s + 3.34e4·24-s − 6.92e4·25-s + 5.24e4·26-s − 1.96e4·27-s + 9.35e4·28-s + ⋯ |
| L(s) = 1 | + 0.484·2-s − 0.577·3-s − 0.764·4-s − 0.336·5-s − 0.279·6-s − 1.05·7-s − 0.855·8-s + 0.333·9-s − 0.163·10-s + 0.383·11-s + 0.441·12-s + 1.20·13-s − 0.510·14-s + 0.194·15-s + 0.349·16-s − 1.36·17-s + 0.161·18-s − 1.45·19-s + 0.257·20-s + 0.608·21-s + 0.186·22-s − 0.397·23-s + 0.494·24-s − 0.886·25-s + 0.584·26-s − 0.192·27-s + 0.805·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.04759963320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04759963320\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 157 | \( 1 + 3.86e6T \) |
| good | 2 | \( 1 - 5.48T + 128T^{2} \) |
| 5 | \( 1 + 94.0T + 7.81e4T^{2} \) |
| 7 | \( 1 + 955.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.69e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 9.55e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.75e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.35e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.31e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.81e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.22e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.82e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.47e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.67e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 3.78e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 7.68e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.84e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.50e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.06e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.89e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.05e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 8.44e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.02e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.58e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.60e6T + 8.07e13T^{2} \) |
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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
−9.827266505653045099327503198771, −8.996717093278274414766949500845, −8.236828596705678415797809010402, −6.64547948404834685731959366198, −6.24390815530785491997295969519, −5.13085645879575621558059042641, −3.96320389589706012457498132858, −3.57579090034972680716408432935, −1.82156361728792112543153158394, −0.090190935413607673685379965548,
0.090190935413607673685379965548, 1.82156361728792112543153158394, 3.57579090034972680716408432935, 3.96320389589706012457498132858, 5.13085645879575621558059042641, 6.24390815530785491997295969519, 6.64547948404834685731959366198, 8.236828596705678415797809010402, 8.996717093278274414766949500845, 9.827266505653045099327503198771
