| L(s) = 1 | − 10.6·3-s + 176.·7-s − 129.·9-s + 400.·11-s − 784.·13-s − 740.·17-s + 2.77e3·19-s − 1.88e3·21-s − 176.·23-s + 3.97e3·27-s + 5.54e3·29-s − 5.73e3·31-s − 4.27e3·33-s − 6.79e3·37-s + 8.36e3·39-s + 1.84e4·41-s − 1.40e4·43-s + 2.33e4·47-s + 1.45e4·49-s + 7.90e3·51-s + 279.·53-s − 2.96e4·57-s − 4.01e4·59-s + 1.05e4·61-s − 2.28e4·63-s − 3.59e4·67-s + 1.88e3·69-s + ⋯ |
| L(s) = 1 | − 0.684·3-s + 1.36·7-s − 0.531·9-s + 0.999·11-s − 1.28·13-s − 0.621·17-s + 1.76·19-s − 0.933·21-s − 0.0697·23-s + 1.04·27-s + 1.22·29-s − 1.07·31-s − 0.683·33-s − 0.816·37-s + 0.881·39-s + 1.71·41-s − 1.16·43-s + 1.54·47-s + 0.863·49-s + 0.425·51-s + 0.0136·53-s − 1.20·57-s − 1.50·59-s + 0.363·61-s − 0.726·63-s − 0.978·67-s + 0.0477·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.875177360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.875177360\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 10.6T + 243T^{2} \) |
| 7 | \( 1 - 176.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 400.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 784.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 740.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.77e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 176.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.54e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.73e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.79e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.84e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.40e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.33e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 279.T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.01e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.05e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.59e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.52e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.70e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.20e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.44e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.02e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.64e4T + 8.58e9T^{2} \) |
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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
−9.449647352828526490609020709849, −8.727238244376127448068244338730, −7.68595256998963869459804178596, −6.99814984989491317155043421870, −5.83949956706987272086841365801, −5.07617907063267436571177447875, −4.38926657598632424543671724143, −2.94761940572418942154916306468, −1.71707970415050604860176163297, −0.66805136388519216015158786221,
0.66805136388519216015158786221, 1.71707970415050604860176163297, 2.94761940572418942154916306468, 4.38926657598632424543671724143, 5.07617907063267436571177447875, 5.83949956706987272086841365801, 6.99814984989491317155043421870, 7.68595256998963869459804178596, 8.727238244376127448068244338730, 9.449647352828526490609020709849
