| L(s) = 1 | − 1.57·3-s − 49·7-s − 240.·9-s + 322.·11-s − 657.·13-s − 665.·17-s − 2.21e3·19-s + 77.3·21-s − 1.58e3·23-s + 763.·27-s + 6.38e3·29-s + 7.71e3·31-s − 509.·33-s − 3.43e3·37-s + 1.03e3·39-s − 6.53e3·41-s − 1.66e4·43-s − 1.25e4·47-s + 2.40e3·49-s + 1.05e3·51-s − 4.24e3·53-s + 3.49e3·57-s + 2.61e4·59-s + 3.26e4·61-s + 1.17e4·63-s + 6.15e3·67-s + 2.50e3·69-s + ⋯ |
| L(s) = 1 | − 0.101·3-s − 0.377·7-s − 0.989·9-s + 0.804·11-s − 1.07·13-s − 0.558·17-s − 1.40·19-s + 0.0382·21-s − 0.625·23-s + 0.201·27-s + 1.41·29-s + 1.44·31-s − 0.0814·33-s − 0.412·37-s + 0.109·39-s − 0.607·41-s − 1.37·43-s − 0.826·47-s + 0.142·49-s + 0.0565·51-s − 0.207·53-s + 0.142·57-s + 0.978·59-s + 1.12·61-s + 0.374·63-s + 0.167·67-s + 0.0633·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.083703171\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.083703171\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 + 1.57T + 243T^{2} \) |
| 11 | \( 1 - 322.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 657.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 665.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.21e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.58e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.38e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.71e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.43e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.53e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.66e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.25e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.24e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.61e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.26e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.15e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.38e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.21e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.52e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.75e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.28e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.66e5T + 8.58e9T^{2} \) |
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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
−9.782284511371474665098828723184, −8.680285587407909733261649271016, −8.206630876619687409254321641679, −6.72777103856625019947596068217, −6.38972268913702554620450519809, −5.12611864228053059723855541455, −4.22491144025317089200886133557, −3.00351130877839183898875788755, −2.05058555101809375536342665869, −0.46735052081396156929527919121,
0.46735052081396156929527919121, 2.05058555101809375536342665869, 3.00351130877839183898875788755, 4.22491144025317089200886133557, 5.12611864228053059723855541455, 6.38972268913702554620450519809, 6.72777103856625019947596068217, 8.206630876619687409254321641679, 8.680285587407909733261649271016, 9.782284511371474665098828723184
