L(s) = 1 | + 8.44·2-s + 39.3·4-s + 36·5-s + 61.9·8-s + 304.·10-s − 295.·11-s − 1.14e3·13-s − 735.·16-s − 1.03e3·17-s + 2.10e3·19-s + 1.41e3·20-s − 2.49e3·22-s + 640.·23-s − 1.82e3·25-s − 9.69e3·26-s − 7.63e3·29-s − 966.·31-s − 8.19e3·32-s − 8.71e3·34-s − 1.77e3·37-s + 1.78e4·38-s + 2.23e3·40-s + 1.19e4·41-s − 1.98e4·43-s − 1.16e4·44-s + 5.41e3·46-s + 2.79e4·47-s + ⋯ |
L(s) = 1 | + 1.49·2-s + 1.22·4-s + 0.643·5-s + 0.342·8-s + 0.961·10-s − 0.736·11-s − 1.88·13-s − 0.718·16-s − 0.866·17-s + 1.33·19-s + 0.791·20-s − 1.09·22-s + 0.252·23-s − 0.585·25-s − 2.81·26-s − 1.68·29-s − 0.180·31-s − 1.41·32-s − 1.29·34-s − 0.212·37-s + 2.00·38-s + 0.220·40-s + 1.11·41-s − 1.63·43-s − 0.905·44-s + 0.377·46-s + 1.84·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 8.44T + 32T^{2} \) |
| 5 | \( 1 - 36T + 3.12e3T^{2} \) |
| 11 | \( 1 + 295.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.14e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.03e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.10e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 640.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.63e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 966.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.77e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.19e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.98e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.79e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.11e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.08e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.38e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.46e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.84e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.52e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.30e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.60e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.53e5T + 8.58e9T^{2} \) |
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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
−9.881979657286512603709250042825, −9.207502939268527660478633566950, −7.61550202510809735817601000809, −6.89163612919673399990815691167, −5.56358338895352989556498758107, −5.22416959608909723891052019174, −4.11143076999694729366494272655, −2.82981369399257176988176842176, −2.08318306735593412294493117859, 0,
2.08318306735593412294493117859, 2.82981369399257176988176842176, 4.11143076999694729366494272655, 5.22416959608909723891052019174, 5.56358338895352989556498758107, 6.89163612919673399990815691167, 7.61550202510809735817601000809, 9.207502939268527660478633566950, 9.881979657286512603709250042825