| L(s) = 1 | − 3.30e3·7-s + 9.26e4·13-s + 4.87e5·19-s − 5.08e4·25-s − 7.68e5·31-s + 9.93e5·37-s − 1.06e7·43-s − 3.34e6·49-s + 4.67e6·61-s − 6.13e7·67-s − 2.30e7·73-s + 5.31e6·79-s − 3.05e8·91-s − 1.03e8·97-s − 5.79e7·103-s + 3.35e8·109-s + 2.39e8·121-s + 127-s + 131-s − 1.61e9·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.37·7-s + 3.24·13-s + 3.73·19-s − 0.130·25-s − 0.831·31-s + 0.530·37-s − 3.12·43-s − 0.579·49-s + 0.337·61-s − 3.04·67-s − 0.811·73-s + 0.136·79-s − 4.46·91-s − 1.16·97-s − 0.514·103-s + 2.38·109-s + 1.11·121-s − 5.14·133-s + 5.88·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.827014570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.827014570\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2032 p^{2} T^{2} + p^{16} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 236 p T + p^{8} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 239294114 T^{2} + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 46304 T + p^{8} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 1793808704 T^{2} + p^{16} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 243664 T + p^{8} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 136092106850 T^{2} + p^{16} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 907364767184 T^{2} + p^{16} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 384164 T + p^{8} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 496982 T + p^{8} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 14953776865664 T^{2} + p^{16} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 5334440 T + p^{8} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 5979540196322 T^{2} + p^{16} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 117184708681040 T^{2} + p^{16} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 138802557649730 T^{2} + p^{16} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2335370 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 30674456 T + p^{8} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 1143138284216930 T^{2} + p^{16} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 11519728 T + p^{8} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 2658244 T + p^{8} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1816636310871074 T^{2} + p^{16} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 6378488973234560 T^{2} + p^{16} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 51595168 T + p^{8} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.61728204227227824036499867211, −11.45193715642960647673325636832, −10.91281036774311043656552634264, −10.07997127270995722134339688496, −9.858639328572820786542640025199, −9.232958438107944026471806957081, −8.824044010145440240759003457909, −8.205406135363580964233790033028, −7.60529141418319556249115862862, −6.94504004015062490039286109158, −6.44073654355030204578362942010, −5.79420676870473664965788702906, −5.57787256994056971618287331131, −4.62182281745277366108373739479, −3.56232397453258706854567682609, −3.28744180268794197744188113856, −3.15633288497077318225601667587, −1.48863628044131504593711786028, −1.31447338591572693760390542893, −0.46637585187285428112383295937,
0.46637585187285428112383295937, 1.31447338591572693760390542893, 1.48863628044131504593711786028, 3.15633288497077318225601667587, 3.28744180268794197744188113856, 3.56232397453258706854567682609, 4.62182281745277366108373739479, 5.57787256994056971618287331131, 5.79420676870473664965788702906, 6.44073654355030204578362942010, 6.94504004015062490039286109158, 7.60529141418319556249115862862, 8.205406135363580964233790033028, 8.824044010145440240759003457909, 9.232958438107944026471806957081, 9.858639328572820786542640025199, 10.07997127270995722134339688496, 10.91281036774311043656552634264, 11.45193715642960647673325636832, 11.61728204227227824036499867211
