| L(s) = 1 | − 8·2-s − 69.1·3-s + 64·4-s + 217.·5-s + 553.·6-s + 343·7-s − 512·8-s + 2.59e3·9-s − 1.73e3·10-s − 4.15e3·11-s − 4.42e3·12-s − 2.19e3·13-s − 2.74e3·14-s − 1.50e4·15-s + 4.09e3·16-s − 2.56e4·17-s − 2.07e4·18-s + 2.38e4·19-s + 1.38e4·20-s − 2.37e4·21-s + 3.32e4·22-s + 9.98e4·23-s + 3.54e4·24-s − 3.10e4·25-s + 1.75e4·26-s − 2.81e4·27-s + 2.19e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.47·3-s + 0.5·4-s + 0.776·5-s + 1.04·6-s + 0.377·7-s − 0.353·8-s + 1.18·9-s − 0.549·10-s − 0.941·11-s − 0.739·12-s − 0.277·13-s − 0.267·14-s − 1.14·15-s + 0.250·16-s − 1.26·17-s − 0.838·18-s + 0.796·19-s + 0.388·20-s − 0.558·21-s + 0.665·22-s + 1.71·23-s + 0.522·24-s − 0.396·25-s + 0.196·26-s − 0.274·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 8T \) |
| 7 | \( 1 - 343T \) |
| 13 | \( 1 + 2.19e3T \) |
| good | 3 | \( 1 + 69.1T + 2.18e3T^{2} \) |
| 5 | \( 1 - 217.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 4.15e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 2.56e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.38e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 9.98e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.03e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.18e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.23e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.30e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.49e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.85e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 4.64e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.39e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.53e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.09e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.07e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.00e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 9.37e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.06e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 7.02e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.79e4T + 8.07e13T^{2} \) |
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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
−10.95607553373297067595925450737, −10.01944773454837551005456478842, −9.039304543343111245192574277871, −7.60794524925975970372725150238, −6.58994411676707160538777872293, −5.59946984990690432897812264878, −4.78806551088656076826388889270, −2.53879734030126855828706317190, −1.16136205844719090728396327703, 0,
1.16136205844719090728396327703, 2.53879734030126855828706317190, 4.78806551088656076826388889270, 5.59946984990690432897812264878, 6.58994411676707160538777872293, 7.60794524925975970372725150238, 9.039304543343111245192574277871, 10.01944773454837551005456478842, 10.95607553373297067595925450737
