| L(s) = 1 | − 149.·2-s + 1.40e4·4-s + 1.56e4·5-s + 3.84e5·7-s − 8.81e5·8-s − 2.33e6·10-s − 6.10e6·11-s + 1.13e6·13-s − 5.73e7·14-s + 1.61e7·16-s + 2.69e7·17-s − 3.29e8·19-s + 2.20e8·20-s + 9.11e8·22-s − 4.70e8·23-s + 2.44e8·25-s − 1.69e8·26-s + 5.41e9·28-s + 4.24e9·29-s + 7.67e9·31-s + 4.81e9·32-s − 4.02e9·34-s + 6.00e9·35-s − 2.83e10·37-s + 4.92e10·38-s − 1.37e10·40-s + 2.11e10·41-s + ⋯ |
| L(s) = 1 | − 1.64·2-s + 1.72·4-s + 0.447·5-s + 1.23·7-s − 1.18·8-s − 0.737·10-s − 1.03·11-s + 0.0651·13-s − 2.03·14-s + 0.240·16-s + 0.271·17-s − 1.60·19-s + 0.769·20-s + 1.71·22-s − 0.662·23-s + 0.199·25-s − 0.107·26-s + 2.12·28-s + 1.32·29-s + 1.55·31-s + 0.792·32-s − 0.447·34-s + 0.552·35-s − 1.81·37-s + 2.65·38-s − 0.531·40-s + 0.694·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - 1.56e4T \) |
| good | 2 | \( 1 + 149.T + 8.19e3T^{2} \) |
| 7 | \( 1 - 3.84e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 6.10e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 1.13e6T + 3.02e14T^{2} \) |
| 17 | \( 1 - 2.69e7T + 9.90e15T^{2} \) |
| 19 | \( 1 + 3.29e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 4.70e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 4.24e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 7.67e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 2.83e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 2.11e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 6.93e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 1.01e11T + 5.46e21T^{2} \) |
| 53 | \( 1 - 1.81e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + 3.10e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 1.76e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 6.25e10T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.13e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.17e11T + 1.67e24T^{2} \) |
| 79 | \( 1 - 3.99e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 8.58e11T + 8.87e24T^{2} \) |
| 89 | \( 1 - 9.67e11T + 2.19e25T^{2} \) |
| 97 | \( 1 - 2.50e12T + 6.73e25T^{2} \) |
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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
−11.96966431872176344558457409124, −10.70231604839450367505876725035, −10.11316360654081564868371575209, −8.524746910734751247905055338606, −8.039472356997977014752522369470, −6.55247109106124805747332622355, −4.85418413571621809445417763957, −2.40429526860341645118354320284, −1.41924993575439483541311374456, 0,
1.41924993575439483541311374456, 2.40429526860341645118354320284, 4.85418413571621809445417763957, 6.55247109106124805747332622355, 8.039472356997977014752522369470, 8.524746910734751247905055338606, 10.11316360654081564868371575209, 10.70231604839450367505876725035, 11.96966431872176344558457409124
