| L(s) = 1 | + 5.65i·2-s − 32.0·4-s + 181. i·5-s − 208.·7-s − 181. i·8-s − 1.02e3·10-s − 2.65e3i·11-s + 877.·13-s − 1.18e3i·14-s + 1.02e3·16-s − 4.42e3i·17-s − 4.19e3·19-s − 5.79e3i·20-s + 1.50e4·22-s + 1.20e4i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s + 1.44i·5-s − 0.608·7-s − 0.353i·8-s − 1.02·10-s − 1.99i·11-s + 0.399·13-s − 0.430i·14-s + 0.250·16-s − 0.901i·17-s − 0.611·19-s − 0.724i·20-s + 1.41·22-s + 0.986i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.287904105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.287904105\) |
| \(L(4)\) |
|
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 | \( 1 - 181. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 208.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 2.65e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 877.T + 4.82e6T^{2} \) |
| 17 | \( 1 + 4.42e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 4.19e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 1.20e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 2.64e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 5.80e3T + 8.87e8T^{2} \) |
| 37 | \( 1 - 4.15e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 7.47e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.46e5T + 6.32e9T^{2} \) |
| 47 | \( 1 - 2.57e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.97e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 3.61e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 2.39e4T + 5.15e10T^{2} \) |
| 67 | \( 1 - 3.53e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 4.96e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.82e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 3.86e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 4.15e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 4.05e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 3.56e5T + 8.32e11T^{2} \) |
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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
−11.46034990397876238273863819095, −10.82984615887264702190680155741, −9.659136360607248396311577287623, −8.540542099034047260110590416913, −7.37576122713492593422965269238, −6.40239535624785582445770139934, −5.70140086635916875428276725584, −3.71905226860592040169608770398, −2.84481774054386746865952822660, −0.45028192833955902962289183329,
1.04745231923377499935517234225, 2.23756229406724430952784104709, 4.11490950673814500029089741822, 4.79452010237209567874056583418, 6.28750977162696349216800777602, 7.85392601439595070832573618365, 8.944821194312433280334627657812, 9.686189540242824406544693864290, 10.64489917392013830236745320178, 12.11847636823027822757531063827
