| L(s) = 1 | − 20.0·2-s − 107.·3-s − 109.·4-s − 2.67e3·5-s + 2.14e3·6-s + 1.98e3·7-s + 1.24e4·8-s − 8.23e3·9-s + 5.36e4·10-s + 3.76e4·11-s + 1.17e4·12-s − 2.51e4·13-s − 3.97e4·14-s + 2.86e5·15-s − 1.93e5·16-s + 1.65e5·18-s + 6.23e5·19-s + 2.93e5·20-s − 2.12e5·21-s − 7.55e5·22-s − 8.89e4·23-s − 1.33e6·24-s + 5.20e6·25-s + 5.04e5·26-s + 2.98e6·27-s − 2.17e5·28-s + 1.16e5·29-s + ⋯ |
| L(s) = 1 | − 0.886·2-s − 0.762·3-s − 0.214·4-s − 1.91·5-s + 0.675·6-s + 0.312·7-s + 1.07·8-s − 0.418·9-s + 1.69·10-s + 0.775·11-s + 0.163·12-s − 0.244·13-s − 0.276·14-s + 1.46·15-s − 0.739·16-s + 0.370·18-s + 1.09·19-s + 0.410·20-s − 0.237·21-s − 0.687·22-s − 0.0662·23-s − 0.820·24-s + 2.66·25-s + 0.216·26-s + 1.08·27-s − 0.0669·28-s + 0.0307·29-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
| \(L(5)\) |
\(\approx\) |
\(0.4054969625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4054969625\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 20.0T + 512T^{2} \) |
| 3 | \( 1 + 107.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.67e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.98e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.76e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 2.51e4T + 1.06e10T^{2} \) |
| 19 | \( 1 - 6.23e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 8.89e4T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.16e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.85e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.75e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.89e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.22e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.58e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.03e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 6.28e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.06e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 9.64e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.78e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.95e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.96e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 7.90e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 8.30e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 2.87e8T + 7.60e17T^{2} \) |
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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
−10.32130474363086261742007142412, −9.107416159249477155416873923138, −8.273645516630816405804177756645, −7.62901723092935688334069462019, −6.65132017483028356871323430147, −5.06291962955363779968910673300, −4.32934315208386105255369630343, −3.21352149567787587139848730949, −1.16436344346492916555651637433, −0.41522262128089139875112432085,
0.41522262128089139875112432085, 1.16436344346492916555651637433, 3.21352149567787587139848730949, 4.32934315208386105255369630343, 5.06291962955363779968910673300, 6.65132017483028356871323430147, 7.62901723092935688334069462019, 8.273645516630816405804177756645, 9.107416159249477155416873923138, 10.32130474363086261742007142412
