| L(s) = 1 | + 38.7·3-s − 496.·5-s − 343·7-s − 684.·9-s + 4.03e3·11-s − 1.44e4·13-s − 1.92e4·15-s − 2.67e4·17-s + 3.66e4·19-s − 1.32e4·21-s − 3.49e4·23-s + 1.68e5·25-s − 1.11e5·27-s + 3.58e4·29-s + 1.98e5·31-s + 1.56e5·33-s + 1.70e5·35-s − 1.14e5·37-s − 5.60e5·39-s + 2.45e5·41-s + 3.02e4·43-s + 3.39e5·45-s − 3.05e5·47-s + 1.17e5·49-s − 1.03e6·51-s − 1.21e6·53-s − 2.00e6·55-s + ⋯ |
| L(s) = 1 | + 0.828·3-s − 1.77·5-s − 0.377·7-s − 0.312·9-s + 0.914·11-s − 1.82·13-s − 1.47·15-s − 1.32·17-s + 1.22·19-s − 0.313·21-s − 0.598·23-s + 2.15·25-s − 1.08·27-s + 0.273·29-s + 1.19·31-s + 0.757·33-s + 0.671·35-s − 0.371·37-s − 1.51·39-s + 0.557·41-s + 0.0579·43-s + 0.555·45-s − 0.428·47-s + 0.142·49-s − 1.09·51-s − 1.11·53-s − 1.62·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 + 343T \) |
| good | 3 | \( 1 - 38.7T + 2.18e3T^{2} \) |
| 5 | \( 1 + 496.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 4.03e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.44e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.67e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.66e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.49e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 3.58e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.98e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.14e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.45e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.02e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 3.05e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.21e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.53e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 4.50e4T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.08e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.69e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.86e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.00e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.28e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 6.50e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.14e7T + 8.07e13T^{2} \) |
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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
−15.05569900760741093739859203013, −14.06701542246417433814737309472, −12.29229899146141975387039684831, −11.47065165084101464662263004385, −9.478803074990492011641460549072, −8.193348197772232574703031973277, −7.08165516485800333005193773450, −4.38166338862504466256642613888, −2.95722235374892833709467497848, 0,
2.95722235374892833709467497848, 4.38166338862504466256642613888, 7.08165516485800333005193773450, 8.193348197772232574703031973277, 9.478803074990492011641460549072, 11.47065165084101464662263004385, 12.29229899146141975387039684831, 14.06701542246417433814737309472, 15.05569900760741093739859203013
