| L(s) = 1 | − 5.87·2-s + 27·3-s − 93.4·4-s + 248.·5-s − 158.·6-s + 1.00e3·7-s + 1.30e3·8-s + 729·9-s − 1.46e3·10-s − 2.52e3·12-s + 3.99e3·13-s − 5.88e3·14-s + 6.72e3·15-s + 4.31e3·16-s − 2.37e4·17-s − 4.28e3·18-s − 4.62e4·19-s − 2.32e4·20-s + 2.70e4·21-s + 2.10e4·23-s + 3.51e4·24-s − 1.61e4·25-s − 2.34e4·26-s + 1.96e4·27-s − 9.36e4·28-s − 1.35e5·29-s − 3.95e4·30-s + ⋯ |
| L(s) = 1 | − 0.519·2-s + 0.577·3-s − 0.730·4-s + 0.890·5-s − 0.299·6-s + 1.10·7-s + 0.898·8-s + 0.333·9-s − 0.462·10-s − 0.421·12-s + 0.504·13-s − 0.573·14-s + 0.514·15-s + 0.263·16-s − 1.17·17-s − 0.173·18-s − 1.54·19-s − 0.650·20-s + 0.637·21-s + 0.361·23-s + 0.518·24-s − 0.207·25-s − 0.262·26-s + 0.192·27-s − 0.806·28-s − 1.03·29-s − 0.267·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 5.87T + 128T^{2} \) |
| 5 | \( 1 - 248.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.00e3T + 8.23e5T^{2} \) |
| 13 | \( 1 - 3.99e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.37e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.62e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.10e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.35e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.20e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.24e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.91e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.89e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 3.23e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.25e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 6.74e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.77e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.64e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.58e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.17e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.34e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.63e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.30e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.36e7T + 8.07e13T^{2} \) |
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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
−9.511030268842707809764614224258, −8.849319310502373191702849269563, −8.249404573005807674771498007633, −7.18386760826784849477249842600, −5.85498185784686951755635378921, −4.75876703409439217870367709531, −3.89787182427082299201538932345, −2.12513516978258412169771024064, −1.52583004737491752374130403236, 0,
1.52583004737491752374130403236, 2.12513516978258412169771024064, 3.89787182427082299201538932345, 4.75876703409439217870367709531, 5.85498185784686951755635378921, 7.18386760826784849477249842600, 8.249404573005807674771498007633, 8.849319310502373191702849269563, 9.511030268842707809764614224258
