L(s) = 1 | − 108.·2-s + 729·3-s + 3.62e3·4-s − 1.56e4·5-s − 7.92e4·6-s − 2.22e5·7-s + 4.96e5·8-s + 5.31e5·9-s + 1.70e6·10-s + 5.89e6·11-s + 2.64e6·12-s + 5.02e6·13-s + 2.42e7·14-s − 1.14e7·15-s − 8.36e7·16-s + 1.53e8·17-s − 5.77e7·18-s − 3.69e8·19-s − 5.68e7·20-s − 1.62e8·21-s − 6.40e8·22-s − 5.84e8·23-s + 3.61e8·24-s − 9.74e8·25-s − 5.46e8·26-s + 3.87e8·27-s − 8.08e8·28-s + ⋯ |
L(s) = 1 | − 1.20·2-s + 0.577·3-s + 0.442·4-s − 0.448·5-s − 0.693·6-s − 0.716·7-s + 0.669·8-s + 0.333·9-s + 0.538·10-s + 1.00·11-s + 0.255·12-s + 0.288·13-s + 0.860·14-s − 0.259·15-s − 1.24·16-s + 1.53·17-s − 0.400·18-s − 1.80·19-s − 0.198·20-s − 0.413·21-s − 1.20·22-s − 0.822·23-s + 0.386·24-s − 0.798·25-s − 0.346·26-s + 0.192·27-s − 0.316·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 729T \) |
| 59 | \( 1 - 4.21e10T \) |
good | 2 | \( 1 + 108.T + 8.19e3T^{2} \) |
| 5 | \( 1 + 1.56e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 2.22e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 5.89e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 5.02e6T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.53e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 3.69e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 5.84e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 4.10e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 3.52e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.57e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 4.03e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 2.10e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 6.19e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 2.85e11T + 2.60e22T^{2} \) |
| 61 | \( 1 + 1.72e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 4.13e10T + 5.48e23T^{2} \) |
| 71 | \( 1 + 6.91e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 1.67e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 9.79e11T + 4.66e24T^{2} \) |
| 83 | \( 1 + 4.95e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 1.48e11T + 2.19e25T^{2} \) |
| 97 | \( 1 - 7.18e11T + 6.73e25T^{2} \) |
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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
−9.799180918070070331720201135128, −8.755193377278841430103614121601, −8.174411164877817295592333983722, −7.15086688996526115367082037743, −6.13032555923607392277491588876, −4.31510935628405947602575639353, −3.51356577481106259898810654862, −2.06919029858374264968213462354, −1.02570391004201724852668406305, 0,
1.02570391004201724852668406305, 2.06919029858374264968213462354, 3.51356577481106259898810654862, 4.31510935628405947602575639353, 6.13032555923607392277491588876, 7.15086688996526115367082037743, 8.174411164877817295592333983722, 8.755193377278841430103614121601, 9.799180918070070331720201135128