| L(s) = 1 | + 5.65i·2-s − 32.0·4-s + 171. i·5-s + 560.·7-s − 181. i·8-s − 967.·10-s − 220. i·11-s + 3.78e3·13-s + 3.17e3i·14-s + 1.02e3·16-s − 2.66e3i·17-s + 1.03e4·19-s − 5.47e3i·20-s + 1.24e3·22-s − 1.47e4i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s + 1.36i·5-s + 1.63·7-s − 0.353i·8-s − 0.967·10-s − 0.165i·11-s + 1.72·13-s + 1.15i·14-s + 0.250·16-s − 0.541i·17-s + 1.50·19-s − 0.684i·20-s + 0.117·22-s − 1.21i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.691159132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.691159132\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 5.65iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 171. iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 560.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 220. iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 3.78e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 2.66e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 1.03e4T + 4.70e7T^{2} \) |
| 23 | \( 1 + 1.47e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 1.48e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 1.58e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 2.65e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 1.13e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 5.20e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.15e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.37e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 2.91e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.82e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 3.77e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 5.21e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 6.11e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 1.75e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 3.62e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 1.23e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.24e6T + 8.32e11T^{2} \) |
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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
−11.59812018178245451010063061039, −11.14852732408904738228958221919, −10.06045139390469356506546478225, −8.571956237611663528654709684587, −7.81230031135916337705648416398, −6.75866893653050673815383371971, −5.71513905087309177936814795036, −4.39816369798921574547822711741, −2.96089421146667115527579611197, −1.19162100874893322881330966975,
1.09952539244804774506001404144, 1.57479375120172598624867436012, 3.69383235514998664572906287737, 4.82981207932974849252419487497, 5.62365526424513542265450551524, 7.77213477856082003324883664468, 8.535408350116776869467345148432, 9.320255421283360437149248052619, 10.76297583165620111512113338760, 11.53367059337650011455805085162
