| L(s) = 1 | − 4-s − 9-s + 16-s − 4·19-s − 6·29-s + 4·31-s + 36-s − 11·49-s − 12·59-s + 4·61-s − 64-s − 30·71-s + 4·76-s − 16·79-s + 81-s − 6·89-s − 6·101-s − 34·109-s + 6·116-s − 22·121-s − 4·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 1/4·16-s − 0.917·19-s − 1.11·29-s + 0.718·31-s + 1/6·36-s − 1.57·49-s − 1.56·59-s + 0.512·61-s − 1/8·64-s − 3.56·71-s + 0.458·76-s − 1.80·79-s + 1/9·81-s − 0.635·89-s − 0.597·101-s − 3.25·109-s + 0.557·116-s − 2·121-s − 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1003897440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1003897440\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 23 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.895639753042030860631324562129, −8.271691042226575472260952390266, −8.228748773206447603987545793015, −7.55530172115355737128113789591, −7.52303623655672188473939267393, −6.90790957916771441219695261284, −6.36008398282496716250461758773, −6.28992031284181088572367084469, −5.79471362257843304807031075609, −5.18458523872596187799710864741, −5.17685949026239054688768788434, −4.46046835061462749639832182058, −4.07143876854910272558375032741, −3.96312841482922974109616866039, −3.03584315091081225000454124690, −2.96681765471800979453144584953, −2.36710823088948219483591087991, −1.52339799204544120641488527494, −1.38228166563826024199385256515, −0.095772021392148762116869379072,
0.095772021392148762116869379072, 1.38228166563826024199385256515, 1.52339799204544120641488527494, 2.36710823088948219483591087991, 2.96681765471800979453144584953, 3.03584315091081225000454124690, 3.96312841482922974109616866039, 4.07143876854910272558375032741, 4.46046835061462749639832182058, 5.17685949026239054688768788434, 5.18458523872596187799710864741, 5.79471362257843304807031075609, 6.28992031284181088572367084469, 6.36008398282496716250461758773, 6.90790957916771441219695261284, 7.52303623655672188473939267393, 7.55530172115355737128113789591, 8.228748773206447603987545793015, 8.271691042226575472260952390266, 8.895639753042030860631324562129
