| L(s) = 1 | + 34.1·2-s + 655.·4-s − 1.42e3·5-s + 2.40e3·7-s + 4.88e3·8-s − 4.86e4·10-s − 6.93e4·11-s + 1.05e5·13-s + 8.20e4·14-s − 1.68e5·16-s − 5.68e5·17-s − 3.96e5·19-s − 9.32e5·20-s − 2.36e6·22-s + 6.20e5·23-s + 7.37e4·25-s + 3.61e6·26-s + 1.57e6·28-s − 4.87e6·29-s − 1.42e6·31-s − 8.25e6·32-s − 1.94e7·34-s − 3.41e6·35-s + 1.31e7·37-s − 1.35e7·38-s − 6.95e6·40-s + 2.03e7·41-s + ⋯ |
| L(s) = 1 | + 1.50·2-s + 1.27·4-s − 1.01·5-s + 0.377·7-s + 0.421·8-s − 1.53·10-s − 1.42·11-s + 1.02·13-s + 0.570·14-s − 0.642·16-s − 1.65·17-s − 0.697·19-s − 1.30·20-s − 2.15·22-s + 0.462·23-s + 0.0377·25-s + 1.55·26-s + 0.483·28-s − 1.28·29-s − 0.277·31-s − 1.39·32-s − 2.49·34-s − 0.385·35-s + 1.14·37-s − 1.05·38-s − 0.429·40-s + 1.12·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 2 | \( 1 - 34.1T + 512T^{2} \) |
| 5 | \( 1 + 1.42e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 6.93e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.05e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 5.68e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.96e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 6.20e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.87e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.42e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.31e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.03e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.11e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.99e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.65e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.09e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 3.20e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 8.02e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.07e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.70e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.16e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.82e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.47e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.09e9T + 7.60e17T^{2} \) |
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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
−12.90893084163688516946747731920, −11.42397333984573649373725490476, −10.93487471758985842688231556022, −8.714252936357873545475765098546, −7.46235458863671044201541615440, −6.02599315580771115065025801236, −4.72079162221499812204051788963, −3.80641240147590263003835847871, −2.38843074118005536910573185663, 0,
2.38843074118005536910573185663, 3.80641240147590263003835847871, 4.72079162221499812204051788963, 6.02599315580771115065025801236, 7.46235458863671044201541615440, 8.714252936357873545475765098546, 10.93487471758985842688231556022, 11.42397333984573649373725490476, 12.90893084163688516946747731920
