| L(s) = 1 | − 3.02·2-s + 27·3-s − 118.·4-s − 81.6·6-s − 1.50e3·7-s + 746.·8-s + 729·9-s + 1.59e3·11-s − 3.20e3·12-s − 956.·13-s + 4.54e3·14-s + 1.29e4·16-s + 3.24e4·17-s − 2.20e3·18-s − 3.91e4·19-s − 4.06e4·21-s − 4.82e3·22-s + 5.93e4·23-s + 2.01e4·24-s + 2.88e3·26-s + 1.96e4·27-s + 1.78e5·28-s + 6.61e4·29-s − 1.96e4·31-s − 1.34e5·32-s + 4.31e4·33-s − 9.81e4·34-s + ⋯ |
| L(s) = 1 | − 0.267·2-s + 0.577·3-s − 0.928·4-s − 0.154·6-s − 1.65·7-s + 0.515·8-s + 0.333·9-s + 0.361·11-s − 0.536·12-s − 0.120·13-s + 0.443·14-s + 0.790·16-s + 1.60·17-s − 0.0890·18-s − 1.31·19-s − 0.957·21-s − 0.0966·22-s + 1.01·23-s + 0.297·24-s + 0.0322·26-s + 0.192·27-s + 1.54·28-s + 0.503·29-s − 0.118·31-s − 0.726·32-s + 0.208·33-s − 0.428·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.285384643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.285384643\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 3.02T + 128T^{2} \) |
| 7 | \( 1 + 1.50e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.59e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 956.T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.24e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.91e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.93e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.61e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.96e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.76e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.85e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.66e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.68e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.60e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.04e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.78e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.64e3T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.79e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.01e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.76e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.06e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 6.14e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.02e6T + 8.07e13T^{2} \) |
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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
−13.03614991956491371217596683548, −12.45312526020632829121850573305, −10.40995589033297398754928463468, −9.567386501450386882363102599012, −8.784418888474675965628471840629, −7.41829928262291887920110907561, −5.96833085965788158847664305618, −4.17457456367466495301318606490, −3.00116470512469888020581603410, −0.77249770179707580692586814446,
0.77249770179707580692586814446, 3.00116470512469888020581603410, 4.17457456367466495301318606490, 5.96833085965788158847664305618, 7.41829928262291887920110907561, 8.784418888474675965628471840629, 9.567386501450386882363102599012, 10.40995589033297398754928463468, 12.45312526020632829121850573305, 13.03614991956491371217596683548
