| L(s) = 1 | − 912. i·5-s − 1.65e3·7-s + 1.37e4i·11-s + 4.63e4·13-s − 1.10e5i·17-s + 2.43e5·19-s − 1.43e5i·23-s − 4.41e5·25-s − 3.05e5i·29-s − 3.84e5·31-s + 1.50e6i·35-s + 4.96e5·37-s + 1.00e6i·41-s − 5.33e6·43-s − 6.45e6i·47-s + ⋯ |
| L(s) = 1 | − 1.45i·5-s − 0.688·7-s + 0.940i·11-s + 1.62·13-s − 1.32i·17-s + 1.86·19-s − 0.512i·23-s − 1.13·25-s − 0.431i·29-s − 0.415·31-s + 1.00i·35-s + 0.265·37-s + 0.356i·41-s − 1.56·43-s − 1.32i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.681372823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.681372823\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 912. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 1.65e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 1.37e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 4.63e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.10e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 2.43e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 1.43e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 3.05e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 3.84e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 4.96e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 1.00e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 5.33e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 6.45e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 2.70e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 1.24e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 2.33e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 3.06e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 1.21e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 1.15e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 2.65e6T + 1.51e15T^{2} \) |
| 83 | \( 1 - 5.18e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 3.86e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 5.15e7T + 7.83e15T^{2} \) |
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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
−11.45193715642960647673325636832, −9.858639328572820786542640025199, −9.232958438107944026471806957081, −8.205406135363580964233790033028, −6.94504004015062490039286109158, −5.57787256994056971618287331131, −4.62182281745277366108373739479, −3.28744180268794197744188113856, −1.48863628044131504593711786028, −0.46637585187285428112383295937,
1.31447338591572693760390542893, 3.15633288497077318225601667587, 3.56232397453258706854567682609, 5.79420676870473664965788702906, 6.44073654355030204578362942010, 7.60529141418319556249115862862, 8.824044010145440240759003457909, 10.07997127270995722134339688496, 10.91281036774311043656552634264, 11.61728204227227824036499867211
