| L(s) = 1 | − 13.6·2-s + 58.8·4-s + 207.·5-s + 882.·7-s + 945.·8-s − 2.83e3·10-s + 3.63e3·11-s + 208.·13-s − 1.20e4·14-s − 2.04e4·16-s + 1.31e4·17-s − 9.91e3·19-s + 1.21e4·20-s − 4.97e4·22-s + 3.77e4·23-s − 3.51e4·25-s − 2.84e3·26-s + 5.19e4·28-s + 9.80e4·29-s + 3.26e5·31-s + 1.58e5·32-s − 1.79e5·34-s + 1.83e5·35-s + 3.94e4·37-s + 1.35e5·38-s + 1.96e5·40-s + 7.19e5·41-s + ⋯ |
| L(s) = 1 | − 1.20·2-s + 0.459·4-s + 0.742·5-s + 0.972·7-s + 0.652·8-s − 0.896·10-s + 0.823·11-s + 0.0262·13-s − 1.17·14-s − 1.24·16-s + 0.647·17-s − 0.331·19-s + 0.340·20-s − 0.995·22-s + 0.647·23-s − 0.449·25-s − 0.0317·26-s + 0.446·28-s + 0.746·29-s + 1.97·31-s + 0.855·32-s − 0.782·34-s + 0.721·35-s + 0.127·37-s + 0.400·38-s + 0.484·40-s + 1.63·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.902221165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.902221165\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 2 | \( 1 + 13.6T + 128T^{2} \) |
| 5 | \( 1 - 207.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 882.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.63e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 208.T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.31e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 9.91e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.77e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 9.80e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 3.26e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.94e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.19e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.78e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.58e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.83e6T + 1.17e12T^{2} \) |
| 61 | \( 1 - 3.49e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.00e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.31e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.27e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.50e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.48e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 6.37e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.33e6T + 8.07e13T^{2} \) |
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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
−9.781320342904411474821560166019, −8.733976479783515106654766899739, −8.242216430595410023159572615717, −7.23398163978442357669756842356, −6.26480846159512422855896109271, −5.08992490189480703747660626873, −4.15149160562539407790997596600, −2.48781692902235961842788090481, −1.44541693781794139674085200592, −0.822318140945129499003997876317,
0.822318140945129499003997876317, 1.44541693781794139674085200592, 2.48781692902235961842788090481, 4.15149160562539407790997596600, 5.08992490189480703747660626873, 6.26480846159512422855896109271, 7.23398163978442357669756842356, 8.242216430595410023159572615717, 8.733976479783515106654766899739, 9.781320342904411474821560166019
