| L(s) = 1 | + 4.36·2-s − 492.·4-s + 2.36e3·5-s + 327.·7-s − 4.38e3·8-s + 1.02e4·10-s − 3.31e4·11-s − 2.85e4·13-s + 1.42e3·14-s + 2.33e5·16-s − 4.45e5·17-s + 2.16e5·19-s − 1.16e6·20-s − 1.44e5·22-s − 1.14e5·23-s + 3.62e6·25-s − 1.24e5·26-s − 1.61e5·28-s + 2.34e6·29-s − 6.82e6·31-s + 3.26e6·32-s − 1.94e6·34-s + 7.73e5·35-s + 1.79e7·37-s + 9.42e5·38-s − 1.03e7·40-s + 1.27e7·41-s + ⋯ |
| L(s) = 1 | + 0.192·2-s − 0.962·4-s + 1.68·5-s + 0.0515·7-s − 0.378·8-s + 0.325·10-s − 0.682·11-s − 0.277·13-s + 0.00994·14-s + 0.889·16-s − 1.29·17-s + 0.380·19-s − 1.62·20-s − 0.131·22-s − 0.0849·23-s + 1.85·25-s − 0.0534·26-s − 0.0496·28-s + 0.615·29-s − 1.32·31-s + 0.549·32-s − 0.249·34-s + 0.0871·35-s + 1.57·37-s + 0.0733·38-s − 0.639·40-s + 0.706·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 2 | \( 1 - 4.36T + 512T^{2} \) |
| 5 | \( 1 - 2.36e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 327.T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.31e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 4.45e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.16e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.14e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.34e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.82e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.79e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.27e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.87e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.76e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.06e8T + 3.29e15T^{2} \) |
| 59 | \( 1 + 4.22e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 6.48e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.26e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.51e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.44e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.28e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.25e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 4.84e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.40e9T + 7.60e17T^{2} \) |
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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
−11.08467733379256769229479391581, −9.884482769091086074040533565475, −9.367095711144714571531557687635, −8.206929767118121897629823952434, −6.53231537896484620113426109172, −5.47938662166482387538530582863, −4.63266806654204210751907254981, −2.87313353945353323704112598878, −1.61301642646559021690241727200, 0,
1.61301642646559021690241727200, 2.87313353945353323704112598878, 4.63266806654204210751907254981, 5.47938662166482387538530582863, 6.53231537896484620113426109172, 8.206929767118121897629823952434, 9.367095711144714571531557687635, 9.884482769091086074040533565475, 11.08467733379256769229479391581
