L(s) = 1 | + 26.6·2-s − 85.9·3-s + 197.·4-s − 1.46e3·5-s − 2.28e3·6-s − 446.·7-s − 8.38e3·8-s − 1.22e4·9-s − 3.88e4·10-s + 2.56e3·11-s − 1.69e4·12-s + 7.04e4·13-s − 1.18e4·14-s + 1.25e5·15-s − 3.24e5·16-s − 8.35e4·17-s − 3.27e5·18-s − 4.74e5·19-s − 2.88e5·20-s + 3.83e4·21-s + 6.83e4·22-s + 1.74e6·23-s + 7.20e5·24-s + 1.80e5·25-s + 1.87e6·26-s + 2.74e6·27-s − 8.80e4·28-s + ⋯ |
L(s) = 1 | + 1.17·2-s − 0.612·3-s + 0.385·4-s − 1.04·5-s − 0.721·6-s − 0.0702·7-s − 0.723·8-s − 0.624·9-s − 1.23·10-s + 0.0528·11-s − 0.236·12-s + 0.684·13-s − 0.0826·14-s + 0.640·15-s − 1.23·16-s − 0.242·17-s − 0.735·18-s − 0.834·19-s − 0.402·20-s + 0.0430·21-s + 0.0621·22-s + 1.30·23-s + 0.443·24-s + 0.0922·25-s + 0.805·26-s + 0.995·27-s − 0.0270·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 + 8.35e4T \) |
good | 2 | \( 1 - 26.6T + 512T^{2} \) |
| 3 | \( 1 + 85.9T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.46e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 446.T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.56e3T + 2.35e9T^{2} \) |
| 13 | \( 1 - 7.04e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 4.74e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.74e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.01e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.10e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 3.74e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.04e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.15e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.43e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.87e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.37e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.21e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.31e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.48e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.45e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.22e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.36e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 9.26e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.44e9T + 7.60e17T^{2} \) |
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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
−15.75165868958560485058326065318, −14.72117055220996862519931652458, −13.26138271627157778527006688744, −12.01023941391589302038039713591, −11.08745444055847505602446487641, −8.601871956367533709247863674306, −6.46223949309428521033859739932, −4.91100464538083217406147490907, −3.41401003838981088558527340527, 0,
3.41401003838981088558527340527, 4.91100464538083217406147490907, 6.46223949309428521033859739932, 8.601871956367533709247863674306, 11.08745444055847505602446487641, 12.01023941391589302038039713591, 13.26138271627157778527006688744, 14.72117055220996862519931652458, 15.75165868958560485058326065318