| L(s) = 1 | − 709.·2-s + 6.56e3·3-s + 3.72e5·4-s − 1.31e6·5-s − 4.65e6·6-s − 5.76e6·7-s − 1.71e8·8-s + 4.30e7·9-s + 9.33e8·10-s + 7.09e8·11-s + 2.44e9·12-s − 3.34e9·13-s + 4.09e9·14-s − 8.62e9·15-s + 7.27e10·16-s − 3.89e10·17-s − 3.05e10·18-s + 5.37e10·19-s − 4.89e11·20-s − 3.78e10·21-s − 5.03e11·22-s − 4.45e11·23-s − 1.12e12·24-s + 9.66e11·25-s + 2.37e12·26-s + 2.82e11·27-s − 2.14e12·28-s + ⋯ |
| L(s) = 1 | − 1.96·2-s + 0.577·3-s + 2.84·4-s − 1.50·5-s − 1.13·6-s − 0.377·7-s − 3.61·8-s + 0.333·9-s + 2.95·10-s + 0.998·11-s + 1.64·12-s − 1.13·13-s + 0.740·14-s − 0.869·15-s + 4.23·16-s − 1.35·17-s − 0.653·18-s + 0.726·19-s − 4.27·20-s − 0.218·21-s − 1.95·22-s − 1.18·23-s − 2.08·24-s + 1.26·25-s + 2.22·26-s + 0.192·27-s − 1.07·28-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
| \(L(9)\) |
\(\approx\) |
\(0.3940436211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3940436211\) |
| \(L(\frac{19}{2})\) |
|
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 | \( 1 + 709.T + 1.31e5T^{2} \) |
| 5 | \( 1 + 1.31e6T + 7.62e11T^{2} \) |
| 11 | \( 1 - 7.09e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 3.34e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 3.89e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 5.37e10T + 5.48e21T^{2} \) |
| 23 | \( 1 + 4.45e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 3.25e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 9.89e10T + 2.25e25T^{2} \) |
| 37 | \( 1 + 1.33e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 2.18e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 4.29e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 9.94e13T + 2.66e28T^{2} \) |
| 53 | \( 1 - 4.58e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 1.36e15T + 1.27e30T^{2} \) |
| 61 | \( 1 - 2.32e14T + 2.24e30T^{2} \) |
| 67 | \( 1 - 4.61e14T + 1.10e31T^{2} \) |
| 71 | \( 1 - 3.27e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 3.54e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.17e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 1.42e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 1.04e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 3.23e16T + 5.95e33T^{2} \) |
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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
−14.86996742801023730489139014915, −12.16225990158517941130633347891, −11.32136781334026420606215299668, −9.792721187359762940180739318641, −8.772602889207988352799455159569, −7.67446262254433134892970815840, −6.82480008178488045278369579289, −3.64266087282884144007497026256, −2.11985317566549272961465607461, −0.46321405534960060931633768836,
0.46321405534960060931633768836, 2.11985317566549272961465607461, 3.64266087282884144007497026256, 6.82480008178488045278369579289, 7.67446262254433134892970815840, 8.772602889207988352799455159569, 9.792721187359762940180739318641, 11.32136781334026420606215299668, 12.16225990158517941130633347891, 14.86996742801023730489139014915
