| L(s) = 1 | + 2·3-s + 2·5-s + 20·7-s − 3·9-s − 80·13-s + 4·15-s + 124·17-s + 72·19-s + 40·21-s + 98·23-s − 55·25-s + 34·27-s − 144·29-s + 34·31-s + 40·35-s + 54·37-s − 160·39-s − 536·41-s − 60·43-s − 6·45-s + 272·47-s − 338·49-s + 248·51-s − 492·53-s + 144·57-s − 634·59-s − 840·61-s + ⋯ |
| L(s) = 1 | + 0.384·3-s + 0.178·5-s + 1.07·7-s − 1/9·9-s − 1.70·13-s + 0.0688·15-s + 1.76·17-s + 0.869·19-s + 0.415·21-s + 0.888·23-s − 0.439·25-s + 0.242·27-s − 0.922·29-s + 0.196·31-s + 0.193·35-s + 0.239·37-s − 0.656·39-s − 2.04·41-s − 0.212·43-s − 0.0198·45-s + 0.844·47-s − 0.985·49-s + 0.680·51-s − 1.27·53-s + 0.334·57-s − 1.39·59-s − 1.76·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3748096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3748096 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.304772998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.304772998\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 7 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 59 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 20 T + 738 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 80 T + 4794 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 124 T + 13238 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 72 T + 4214 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 98 T + 22847 T^{2} - 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 144 T + 44554 T^{2} + 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 34 T + 57519 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 54 T + 101843 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 536 T + 209618 T^{2} + 536 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 60 T + 159146 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 272 T + 182942 T^{2} - 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 492 T + 348862 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 634 T + 458975 T^{2} + 634 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 840 T + 528794 T^{2} + 840 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 754 T + 742455 T^{2} + 754 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 678 T + 813415 T^{2} - 678 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 400 T + 160962 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 p T - 279966 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 468 T + 1155130 T^{2} - 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1842 T + 1935427 T^{2} + 1842 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2194 T + 2966547 T^{2} - 2194 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.073478652187552645013872380145, −8.668713830052756996166413469618, −8.069314004067824914806206309027, −7.80878109021053799219106734312, −7.48604242322138054949041497653, −7.45080514712445211517017587853, −6.57455903811688613081368899303, −6.43763001295170525340536432752, −5.54575749685884778098253193593, −5.45427272135073727085918762021, −4.86113263282202880941353226033, −4.84327047937048172837008824224, −4.15727169933013926277915421826, −3.46855031830109943323308114556, −2.97455361230639991442361831714, −2.91515309100638842056793562219, −1.96215967843596157881545104610, −1.62394600337126027494968317020, −1.17092688024732954744769123660, −0.30415693869859700859173998442,
0.30415693869859700859173998442, 1.17092688024732954744769123660, 1.62394600337126027494968317020, 1.96215967843596157881545104610, 2.91515309100638842056793562219, 2.97455361230639991442361831714, 3.46855031830109943323308114556, 4.15727169933013926277915421826, 4.84327047937048172837008824224, 4.86113263282202880941353226033, 5.45427272135073727085918762021, 5.54575749685884778098253193593, 6.43763001295170525340536432752, 6.57455903811688613081368899303, 7.45080514712445211517017587853, 7.48604242322138054949041497653, 7.80878109021053799219106734312, 8.069314004067824914806206309027, 8.668713830052756996166413469618, 9.073478652187552645013872380145
