| L(s) = 1 | − 649.·2-s − 6.56e3·3-s + 2.91e5·4-s + 5.08e5·5-s + 4.26e6·6-s − 5.76e6·7-s − 1.04e8·8-s + 4.30e7·9-s − 3.30e8·10-s − 9.87e8·11-s − 1.91e9·12-s + 1.13e9·13-s + 3.74e9·14-s − 3.33e9·15-s + 2.95e10·16-s + 3.74e9·17-s − 2.79e10·18-s + 7.42e10·19-s + 1.48e11·20-s + 3.78e10·21-s + 6.41e11·22-s + 2.51e11·23-s + 6.83e11·24-s − 5.04e11·25-s − 7.40e11·26-s − 2.82e11·27-s − 1.67e12·28-s + ⋯ |
| L(s) = 1 | − 1.79·2-s − 0.577·3-s + 2.22·4-s + 0.581·5-s + 1.03·6-s − 0.377·7-s − 2.19·8-s + 0.333·9-s − 1.04·10-s − 1.38·11-s − 1.28·12-s + 0.387·13-s + 0.678·14-s − 0.335·15-s + 1.71·16-s + 0.130·17-s − 0.598·18-s + 1.00·19-s + 1.29·20-s + 0.218·21-s + 2.49·22-s + 0.669·23-s + 1.26·24-s − 0.661·25-s − 0.695·26-s − 0.192·27-s − 0.840·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 6.56e3T \) |
| 7 | \( 1 + 5.76e6T \) |
| good | 2 | \( 1 + 649.T + 1.31e5T^{2} \) |
| 5 | \( 1 - 5.08e5T + 7.62e11T^{2} \) |
| 11 | \( 1 + 9.87e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 1.13e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 3.74e9T + 8.27e20T^{2} \) |
| 19 | \( 1 - 7.42e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 2.51e11T + 1.41e23T^{2} \) |
| 29 | \( 1 - 5.43e11T + 7.25e24T^{2} \) |
| 31 | \( 1 - 4.88e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 6.69e12T + 4.56e26T^{2} \) |
| 41 | \( 1 + 9.55e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 1.18e14T + 5.87e27T^{2} \) |
| 47 | \( 1 + 2.80e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 7.83e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 1.50e14T + 1.27e30T^{2} \) |
| 61 | \( 1 - 2.12e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 1.86e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 7.74e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 1.75e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 1.79e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 2.17e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 9.77e15T + 1.37e33T^{2} \) |
| 97 | \( 1 + 7.76e16T + 5.95e33T^{2} \) |
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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
−13.30614740818663621028427778676, −11.65557500677723759476307192999, −10.42641771881263027874147244944, −9.692143425138789535744797493617, −8.218035956531980095727388905303, −6.94160379091003956903370026138, −5.56673738989229546085716359193, −2.68337861231909245995073417418, −1.21158503784375269817710950920, 0,
1.21158503784375269817710950920, 2.68337861231909245995073417418, 5.56673738989229546085716359193, 6.94160379091003956903370026138, 8.218035956531980095727388905303, 9.692143425138789535744797493617, 10.42641771881263027874147244944, 11.65557500677723759476307192999, 13.30614740818663621028427778676
