| L(s) = 1 | − 40.6·2-s + 102.·3-s + 1.13e3·4-s + 769.·5-s − 4.18e3·6-s − 1.56e3·7-s − 2.53e4·8-s − 9.08e3·9-s − 3.12e4·10-s + 1.70e4·11-s + 1.17e5·12-s + 1.41e5·13-s + 6.36e4·14-s + 7.92e4·15-s + 4.48e5·16-s + 3.68e5·18-s − 7.43e5·19-s + 8.75e5·20-s − 1.61e5·21-s − 6.91e5·22-s + 1.55e6·23-s − 2.61e6·24-s − 1.36e6·25-s − 5.76e6·26-s − 2.96e6·27-s − 1.78e6·28-s − 5.73e6·29-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 0.733·3-s + 2.22·4-s + 0.550·5-s − 1.31·6-s − 0.246·7-s − 2.19·8-s − 0.461·9-s − 0.988·10-s + 0.350·11-s + 1.63·12-s + 1.37·13-s + 0.442·14-s + 0.404·15-s + 1.71·16-s + 0.828·18-s − 1.30·19-s + 1.22·20-s − 0.181·21-s − 0.629·22-s + 1.15·23-s − 1.60·24-s − 0.696·25-s − 2.47·26-s − 1.07·27-s − 0.547·28-s − 1.50·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.234663309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.234663309\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 40.6T + 512T^{2} \) |
| 3 | \( 1 - 102.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 769.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.56e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 1.70e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.41e5T + 1.06e10T^{2} \) |
| 19 | \( 1 + 7.43e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.55e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.73e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.58e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 5.48e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.81e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.79e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.65e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 7.58e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 4.75e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.40e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 7.10e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 7.14e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.40e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.20e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.26e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.75e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 6.65e8T + 7.60e17T^{2} \) |
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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
−9.929795865635895358861354065062, −9.034208423565548276462307349687, −8.672907385423214349321899883732, −7.76027960121979108650146353561, −6.61786313683342146475884205612, −5.84767256432085155918487899080, −3.73273117109840060909372459136, −2.53961333485675050608611572192, −1.71431033548214547206467679494, −0.62559511185732605878051928389,
0.62559511185732605878051928389, 1.71431033548214547206467679494, 2.53961333485675050608611572192, 3.73273117109840060909372459136, 5.84767256432085155918487899080, 6.61786313683342146475884205612, 7.76027960121979108650146353561, 8.672907385423214349321899883732, 9.034208423565548276462307349687, 9.929795865635895358861354065062
