| L(s) = 1 | − 27.3·3-s − 52.2·5-s + 49·7-s + 503.·9-s − 142.·11-s − 219.·13-s + 1.42e3·15-s − 1.55e3·17-s − 372.·19-s − 1.33e3·21-s + 3.65e3·23-s − 397.·25-s − 7.11e3·27-s − 4.49e3·29-s − 9.15e3·31-s + 3.88e3·33-s − 2.55e3·35-s − 3.66e3·37-s + 5.99e3·39-s − 1.76e4·41-s − 1.10e4·43-s − 2.62e4·45-s − 1.97e4·47-s + 2.40e3·49-s + 4.25e4·51-s − 2.29e4·53-s + 7.43e3·55-s + ⋯ |
| L(s) = 1 | − 1.75·3-s − 0.934·5-s + 0.377·7-s + 2.07·9-s − 0.354·11-s − 0.359·13-s + 1.63·15-s − 1.30·17-s − 0.236·19-s − 0.662·21-s + 1.44·23-s − 0.127·25-s − 1.87·27-s − 0.991·29-s − 1.71·31-s + 0.621·33-s − 0.353·35-s − 0.440·37-s + 0.630·39-s − 1.64·41-s − 0.914·43-s − 1.93·45-s − 1.30·47-s + 0.142·49-s + 2.28·51-s − 1.12·53-s + 0.331·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2166325361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2166325361\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 + 27.3T + 243T^{2} \) |
| 5 | \( 1 + 52.2T + 3.12e3T^{2} \) |
| 11 | \( 1 + 142.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 219.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.55e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 372.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.65e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.49e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.66e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.76e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.10e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.97e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.29e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.06e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.32e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.49e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.66e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.84e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.58e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.61e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.82e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.23e5T + 8.58e9T^{2} \) |
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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
−10.78366614594710752031897046664, −9.588643650389414504833212264481, −8.355794516517505700643938175845, −7.21393585128463028594081387728, −6.66410810100550033932400971570, −5.33682623868949842028110751203, −4.81289676803178302247069072559, −3.69535927691362720837823028085, −1.75665894473032280656439790442, −0.25953265639030810536433321192,
0.25953265639030810536433321192, 1.75665894473032280656439790442, 3.69535927691362720837823028085, 4.81289676803178302247069072559, 5.33682623868949842028110751203, 6.66410810100550033932400971570, 7.21393585128463028594081387728, 8.355794516517505700643938175845, 9.588643650389414504833212264481, 10.78366614594710752031897046664
