| L(s) = 1 | + 26.5·2-s + 81·3-s + 190.·4-s + 1.29e3·5-s + 2.14e3·6-s − 3.38e3·7-s − 8.52e3·8-s + 6.56e3·9-s + 3.43e4·10-s + 8.29e4·11-s + 1.54e4·12-s + 1.56e5·13-s − 8.97e4·14-s + 1.05e5·15-s − 3.23e5·16-s + 1.22e5·17-s + 1.73e5·18-s − 1.30e5·19-s + 2.47e5·20-s − 2.74e5·21-s + 2.19e6·22-s + 1.72e6·23-s − 6.90e5·24-s − 2.71e5·25-s + 4.13e6·26-s + 5.31e5·27-s − 6.44e5·28-s + ⋯ |
| L(s) = 1 | + 1.17·2-s + 0.577·3-s + 0.372·4-s + 0.928·5-s + 0.676·6-s − 0.532·7-s − 0.735·8-s + 0.333·9-s + 1.08·10-s + 1.70·11-s + 0.214·12-s + 1.51·13-s − 0.624·14-s + 0.535·15-s − 1.23·16-s + 0.355·17-s + 0.390·18-s − 0.229·19-s + 0.345·20-s − 0.307·21-s + 2.00·22-s + 1.28·23-s − 0.424·24-s − 0.138·25-s + 1.77·26-s + 0.192·27-s − 0.198·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(5.144575876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.144575876\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 19 | \( 1 + 1.30e5T \) |
| good | 2 | \( 1 - 26.5T + 512T^{2} \) |
| 5 | \( 1 - 1.29e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 3.38e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 8.29e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.56e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.22e5T + 1.18e11T^{2} \) |
| 23 | \( 1 - 1.72e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.25e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.93e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 6.00e5T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.80e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.16e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 8.08e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.67e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 9.71e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.59e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.71e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.38e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 7.79e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.96e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 7.10e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.04e9T + 3.50e17T^{2} \) |
| 97 | \( 1 - 3.58e8T + 7.60e17T^{2} \) |
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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
−13.44147254156790786017096318296, −12.63519953212839916704914774145, −11.25375130056020728367951185724, −9.529606262605644189790777064501, −8.837884647736523485960891868210, −6.64361359829163123598492083350, −5.82301459918773624724813704423, −4.12659123642001577064196468572, −3.13822252147872819712721812270, −1.40469625205078531351679646561,
1.40469625205078531351679646561, 3.13822252147872819712721812270, 4.12659123642001577064196468572, 5.82301459918773624724813704423, 6.64361359829163123598492083350, 8.837884647736523485960891868210, 9.529606262605644189790777064501, 11.25375130056020728367951185724, 12.63519953212839916704914774145, 13.44147254156790786017096318296
