| L(s) = 1 | + 41.7·2-s − 81·3-s + 1.23e3·4-s − 625·5-s − 3.38e3·6-s + 2.40e3·7-s + 3.01e4·8-s + 6.56e3·9-s − 2.61e4·10-s − 1.90e4·11-s − 9.98e4·12-s + 1.72e5·13-s + 1.00e5·14-s + 5.06e4·15-s + 6.26e5·16-s − 2.53e5·17-s + 2.74e5·18-s + 4.04e4·19-s − 7.70e5·20-s − 1.94e5·21-s − 7.96e5·22-s + 1.52e6·23-s − 2.43e6·24-s + 3.90e5·25-s + 7.18e6·26-s − 5.31e5·27-s + 2.95e6·28-s + ⋯ |
| L(s) = 1 | + 1.84·2-s − 0.577·3-s + 2.40·4-s − 0.447·5-s − 1.06·6-s + 0.377·7-s + 2.59·8-s + 0.333·9-s − 0.825·10-s − 0.392·11-s − 1.39·12-s + 1.67·13-s + 0.697·14-s + 0.258·15-s + 2.38·16-s − 0.736·17-s + 0.615·18-s + 0.0712·19-s − 1.07·20-s − 0.218·21-s − 0.724·22-s + 1.13·23-s − 1.50·24-s + 0.200·25-s + 3.08·26-s − 0.192·27-s + 0.909·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(6.125176370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.125176370\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 5 | \( 1 + 625T \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 2 | \( 1 - 41.7T + 512T^{2} \) |
| 11 | \( 1 + 1.90e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.72e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.53e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 4.04e4T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.52e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.68e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.33e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 8.76e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.45e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.34e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.96e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.41e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.51e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 5.29e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.37e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.28e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.17e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.26e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.45e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.80e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 3.85e8T + 7.60e17T^{2} \) |
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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
−12.04904719233437338036022007707, −11.27734275187878922148482597356, −10.60148964297346978706610069797, −8.393849210492640090775587799354, −6.95186532724292807329950040811, −6.07556134509474513744922988271, −4.94353969155180190866730386050, −4.06881969567614597666058745167, −2.81707418831635136005104752918, −1.17551097781652903446139097814,
1.17551097781652903446139097814, 2.81707418831635136005104752918, 4.06881969567614597666058745167, 4.94353969155180190866730386050, 6.07556134509474513744922988271, 6.95186532724292807329950040811, 8.393849210492640090775587799354, 10.60148964297346978706610069797, 11.27734275187878922148482597356, 12.04904719233437338036022007707
