| L(s) = 1 | + 6.54·3-s + 45.9·5-s − 200.·9-s + 551.·11-s − 1.09e3·13-s + 301.·15-s + 1.18e3·17-s − 1.16e3·19-s − 44.3·23-s − 1.00e3·25-s − 2.90e3·27-s + 3.32e3·29-s + 8.78e3·31-s + 3.61e3·33-s − 2.55e3·37-s − 7.16e3·39-s + 1.27e4·41-s + 96.7·43-s − 9.20e3·45-s + 7.67e3·47-s + 7.73e3·51-s − 1.19e4·53-s + 2.53e4·55-s − 7.63e3·57-s + 9.85e3·59-s + 3.85e4·61-s − 5.03e4·65-s + ⋯ |
| L(s) = 1 | + 0.420·3-s + 0.822·5-s − 0.823·9-s + 1.37·11-s − 1.79·13-s + 0.345·15-s + 0.990·17-s − 0.741·19-s − 0.0174·23-s − 0.323·25-s − 0.765·27-s + 0.735·29-s + 1.64·31-s + 0.577·33-s − 0.307·37-s − 0.754·39-s + 1.18·41-s + 0.00798·43-s − 0.677·45-s + 0.507·47-s + 0.416·51-s − 0.584·53-s + 1.13·55-s − 0.311·57-s + 0.368·59-s + 1.32·61-s − 1.47·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.900328567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.900328567\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 6.54T + 243T^{2} \) |
| 5 | \( 1 - 45.9T + 3.12e3T^{2} \) |
| 11 | \( 1 - 551.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.09e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.18e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.16e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 44.3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.78e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.55e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.27e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 96.7T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.67e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.19e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 9.85e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.85e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.75e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.85e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.52T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.50e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.36e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.11e3T + 8.58e9T^{2} \) |
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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
−9.732421478820730212376951867332, −8.748386940421399161693499859284, −7.943953109677436820828500985617, −6.86457318912542789871683020847, −6.06236846835353797851874477646, −5.14786779563350748363763830657, −4.05516040807204652221302834739, −2.81950507290415785181876466756, −2.07793518171309791100860104357, −0.75787750129704031880725959723,
0.75787750129704031880725959723, 2.07793518171309791100860104357, 2.81950507290415785181876466756, 4.05516040807204652221302834739, 5.14786779563350748363763830657, 6.06236846835353797851874477646, 6.86457318912542789871683020847, 7.943953109677436820828500985617, 8.748386940421399161693499859284, 9.732421478820730212376951867332
