| L(s) = 1 | − 64·2-s − 599.·3-s + 4.09e3·4-s + 3.96e4·5-s + 3.83e4·6-s + 3.34e5·7-s − 2.62e5·8-s − 1.23e6·9-s − 2.53e6·10-s + 1.77e6·11-s − 2.45e6·12-s − 5.79e6·13-s − 2.13e7·14-s − 2.37e7·15-s + 1.67e7·16-s − 1.34e7·17-s + 7.90e7·18-s + 5.59e7·19-s + 1.62e8·20-s − 2.00e8·21-s − 1.13e8·22-s + 3.72e8·23-s + 1.57e8·24-s + 3.49e8·25-s + 3.70e8·26-s + 1.69e9·27-s + 1.36e9·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.474·3-s + 0.5·4-s + 1.13·5-s + 0.335·6-s + 1.07·7-s − 0.353·8-s − 0.774·9-s − 0.801·10-s + 0.301·11-s − 0.237·12-s − 0.332·13-s − 0.758·14-s − 0.538·15-s + 0.250·16-s − 0.135·17-s + 0.547·18-s + 0.272·19-s + 0.567·20-s − 0.509·21-s − 0.213·22-s + 0.525·23-s + 0.167·24-s + 0.286·25-s + 0.235·26-s + 0.842·27-s + 0.536·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.559722933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.559722933\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 64T \) |
| 11 | \( 1 - 1.77e6T \) |
| good | 3 | \( 1 + 599.T + 1.59e6T^{2} \) |
| 5 | \( 1 - 3.96e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 3.34e5T + 9.68e10T^{2} \) |
| 13 | \( 1 + 5.79e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.34e7T + 9.90e15T^{2} \) |
| 19 | \( 1 - 5.59e7T + 4.20e16T^{2} \) |
| 23 | \( 1 - 3.72e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 1.55e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 6.94e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.45e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 3.23e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 3.77e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 4.52e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 2.65e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + 3.39e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 3.30e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 1.70e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 4.54e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 2.56e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 3.03e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 4.44e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 2.31e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 1.12e13T + 6.73e25T^{2} \) |
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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
−14.87474104117674250529636473941, −13.76683523600346025627041116867, −11.91873527139603806770244825387, −10.85705119327016153745744460672, −9.531570423869047221233490692268, −8.185503035538439615815396270225, −6.39464833743860959886997294105, −5.11282740790511666289200509508, −2.40424951975807663527369403340, −0.988929169996361016477895706346,
0.988929169996361016477895706346, 2.40424951975807663527369403340, 5.11282740790511666289200509508, 6.39464833743860959886997294105, 8.185503035538439615815396270225, 9.531570423869047221233490692268, 10.85705119327016153745744460672, 11.91873527139603806770244825387, 13.76683523600346025627041116867, 14.87474104117674250529636473941
