| L(s) = 1 | + 2.83e4·5-s − 1.20e5·7-s + 2.03e6·11-s − 6.77e6·13-s − 4.68e7·17-s + 2.37e8·19-s − 3.19e8·23-s − 4.18e8·25-s − 6.56e8·29-s + 2.79e7·31-s − 3.40e9·35-s + 1.45e10·37-s − 2.93e10·41-s − 8.73e9·43-s − 5.43e10·47-s − 8.24e10·49-s − 6.14e10·53-s + 5.76e10·55-s + 2.17e11·59-s + 3.22e10·61-s − 1.91e11·65-s − 3.18e11·67-s + 1.22e12·71-s + 7.02e11·73-s − 2.44e11·77-s + 5.01e11·79-s + 3.38e12·83-s + ⋯ |
| L(s) = 1 | + 0.810·5-s − 0.385·7-s + 0.346·11-s − 0.389·13-s − 0.470·17-s + 1.15·19-s − 0.450·23-s − 0.342·25-s − 0.204·29-s + 0.00565·31-s − 0.312·35-s + 0.933·37-s − 0.965·41-s − 0.210·43-s − 0.735·47-s − 0.851·49-s − 0.380·53-s + 0.280·55-s + 0.669·59-s + 0.0800·61-s − 0.315·65-s − 0.430·67-s + 1.13·71-s + 0.542·73-s − 0.133·77-s + 0.232·79-s + 1.13·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.83e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 1.20e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 2.03e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 6.77e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + 4.68e7T + 9.90e15T^{2} \) |
| 19 | \( 1 - 2.37e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 3.19e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 6.56e8T + 1.02e19T^{2} \) |
| 31 | \( 1 - 2.79e7T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.45e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 2.93e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 8.73e9T + 1.71e21T^{2} \) |
| 47 | \( 1 + 5.43e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 6.14e10T + 2.60e22T^{2} \) |
| 59 | \( 1 - 2.17e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 3.22e10T + 1.61e23T^{2} \) |
| 67 | \( 1 + 3.18e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.22e12T + 1.16e24T^{2} \) |
| 73 | \( 1 - 7.02e11T + 1.67e24T^{2} \) |
| 79 | \( 1 - 5.01e11T + 4.66e24T^{2} \) |
| 83 | \( 1 - 3.38e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 6.98e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 5.81e12T + 6.73e25T^{2} \) |
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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
−9.876695313596523912679744123402, −9.462942123663831972162957424873, −8.135583169701943750355025648623, −6.91842396950592997474636140870, −5.98683452675024230565652839837, −4.95199912564355108430558343970, −3.59554736805315212337257819713, −2.40945065489638487046420854691, −1.34362319930844947134167903687, 0,
1.34362319930844947134167903687, 2.40945065489638487046420854691, 3.59554736805315212337257819713, 4.95199912564355108430558343970, 5.98683452675024230565652839837, 6.91842396950592997474636140870, 8.135583169701943750355025648623, 9.462942123663831972162957424873, 9.876695313596523912679744123402
