| L(s) = 1 | − 5.19i·3-s + 42·5-s − 76.2i·7-s − 27·9-s − 20.7i·11-s + 182·13-s − 218. i·15-s − 246·17-s − 117. i·19-s − 396·21-s + 748. i·23-s + 1.13e3·25-s + 140. i·27-s − 78·29-s − 1.47e3i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.67·5-s − 1.55i·7-s − 0.333·9-s − 0.171i·11-s + 1.07·13-s − 0.969i·15-s − 0.851·17-s − 0.326i·19-s − 0.897·21-s + 1.41i·23-s + 1.82·25-s + 0.192i·27-s − 0.0927·29-s − 1.53i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.456277875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.456277875\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 5.19iT \) |
| good | 5 | \( 1 - 42T + 625T^{2} \) |
| 7 | \( 1 + 76.2iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 20.7iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 182T + 2.85e4T^{2} \) |
| 17 | \( 1 + 246T + 8.35e4T^{2} \) |
| 19 | \( 1 + 117. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 748. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 78T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.47e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 530T + 1.87e6T^{2} \) |
| 41 | \( 1 + 918T + 2.82e6T^{2} \) |
| 43 | \( 1 + 852. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 3.78e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 4.62e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 228. iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 1.34e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 1.08e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 1.82e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 926T + 2.83e7T^{2} \) |
| 79 | \( 1 - 4.39e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 1.19e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.15e4T + 6.27e7T^{2} \) |
| 97 | \( 1 + 1.31e4T + 8.85e7T^{2} \) |
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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
−11.44869314188011670224108434895, −10.58544026464937081782700756380, −9.726960182644347710969659176516, −8.660633982726283366491846776974, −7.27533644260360429654371665584, −6.43757257973687223372246432545, −5.43583839228069757058869336980, −3.78998382436171677216514922442, −2.04714792667969420759364680045, −0.913792393046645537420334612113,
1.80115705674542605631928302001, 2.87635204302111670242438518160, 4.80960687606415568008633795096, 5.83282924104122986019724831100, 6.44921764496777643039157612605, 8.687702816114288192447939600830, 8.975716046987329542309371260056, 10.09606577228384718873383102606, 10.90523813072270801231993214845, 12.19555631875313904409188874271
