| L(s) = 1 | − 25.1·2-s + 81·3-s + 121.·4-s − 1.28e3·5-s − 2.03e3·6-s + 9.83e3·8-s + 6.56e3·9-s + 3.23e4·10-s + 4.32e4·11-s + 9.80e3·12-s + 1.89e5·13-s − 1.04e5·15-s − 3.09e5·16-s − 3.02e5·17-s − 1.65e5·18-s + 2.19e5·19-s − 1.55e5·20-s − 1.08e6·22-s − 1.26e6·23-s + 7.96e5·24-s − 3.03e5·25-s − 4.76e6·26-s + 5.31e5·27-s + 5.46e6·29-s + 2.61e6·30-s + 1.70e5·31-s + 2.74e6·32-s + ⋯ |
| L(s) = 1 | − 1.11·2-s + 0.577·3-s + 0.236·4-s − 0.918·5-s − 0.641·6-s + 0.849·8-s + 0.333·9-s + 1.02·10-s + 0.891·11-s + 0.136·12-s + 1.83·13-s − 0.530·15-s − 1.18·16-s − 0.879·17-s − 0.370·18-s + 0.385·19-s − 0.217·20-s − 0.990·22-s − 0.940·23-s + 0.490·24-s − 0.155·25-s − 2.04·26-s + 0.192·27-s + 1.43·29-s + 0.589·30-s + 0.0331·31-s + 0.463·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.183535321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.183535321\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 25.1T + 512T^{2} \) |
| 5 | \( 1 + 1.28e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 4.32e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.89e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.02e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.19e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.26e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.46e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.70e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.43e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.00e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.37e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.09e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.24e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 8.01e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 6.98e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.30e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.30e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.83e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.43e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.33e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 6.57e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.14e9T + 7.60e17T^{2} \) |
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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
−11.12037399949453233832008868236, −10.17346720997537493639028867768, −8.884216169420088335265166856125, −8.541995659308207256464191980582, −7.52738410799738245832316027202, −6.38989708358075656253405260060, −4.37584609228199000316089790231, −3.56037546438999035231518220939, −1.72531201453154949032151724970, −0.67763999676329004632203523535,
0.67763999676329004632203523535, 1.72531201453154949032151724970, 3.56037546438999035231518220939, 4.37584609228199000316089790231, 6.38989708358075656253405260060, 7.52738410799738245832316027202, 8.541995659308207256464191980582, 8.884216169420088335265166856125, 10.17346720997537493639028867768, 11.12037399949453233832008868236
