| L(s) = 1 | + 8.52·2-s + 40.6·4-s − 160.·7-s + 73.3·8-s − 279.·11-s + 541.·13-s − 1.36e3·14-s − 674.·16-s + 777.·17-s − 2.68e3·19-s − 2.38e3·22-s − 3.69e3·23-s + 4.61e3·26-s − 6.51e3·28-s − 8.35e3·29-s − 262.·31-s − 8.09e3·32-s + 6.62e3·34-s + 1.49e4·37-s − 2.28e4·38-s + 7.98e3·41-s − 5.13e3·43-s − 1.13e4·44-s − 3.14e4·46-s + 1.05e4·47-s + 8.92e3·49-s + 2.19e4·52-s + ⋯ |
| L(s) = 1 | + 1.50·2-s + 1.26·4-s − 1.23·7-s + 0.404·8-s − 0.696·11-s + 0.888·13-s − 1.86·14-s − 0.658·16-s + 0.652·17-s − 1.70·19-s − 1.04·22-s − 1.45·23-s + 1.33·26-s − 1.57·28-s − 1.84·29-s − 0.0491·31-s − 1.39·32-s + 0.982·34-s + 1.79·37-s − 2.56·38-s + 0.742·41-s − 0.423·43-s − 0.883·44-s − 2.19·46-s + 0.697·47-s + 0.531·49-s + 1.12·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 8.52T + 32T^{2} \) |
| 7 | \( 1 + 160.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 279.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 541.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 777.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.68e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.69e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 8.35e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 262.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.49e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.98e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.13e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.10e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.56e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.19e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.19e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.36e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.09e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.92e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.59e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.40e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.22e4T + 8.58e9T^{2} \) |
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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
−11.13764399397377995224614066235, −10.14324719965026221527720846814, −8.964894260383938287095586866957, −7.57952566306347684415192736377, −6.17445127414103162941265437063, −5.87045091493323521346934829111, −4.29455438392342282520932618417, −3.48414062383931890460941667296, −2.29749144766252873352355448319, 0,
2.29749144766252873352355448319, 3.48414062383931890460941667296, 4.29455438392342282520932618417, 5.87045091493323521346934829111, 6.17445127414103162941265437063, 7.57952566306347684415192736377, 8.964894260383938287095586866957, 10.14324719965026221527720846814, 11.13764399397377995224614066235
