| L(s) = 1 | − 4-s + 2·9-s + 8·11-s + 16-s + 10·19-s − 20·29-s − 8·31-s − 2·36-s + 14·41-s − 8·44-s + 14·49-s + 22·59-s + 4·61-s − 64-s + 28·71-s − 10·76-s + 22·79-s − 5·81-s + 4·89-s + 16·99-s − 18·101-s + 32·109-s + 20·116-s + 26·121-s + 8·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2/3·9-s + 2.41·11-s + 1/4·16-s + 2.29·19-s − 3.71·29-s − 1.43·31-s − 1/3·36-s + 2.18·41-s − 1.20·44-s + 2·49-s + 2.86·59-s + 0.512·61-s − 1/8·64-s + 3.32·71-s − 1.14·76-s + 2.47·79-s − 5/9·81-s + 0.423·89-s + 1.60·99-s − 1.79·101-s + 3.06·109-s + 1.85·116-s + 2.36·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.182132617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.182132617\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 37 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 137 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.353836511598731687283558502368, −9.162980863271723136730212827289, −8.964739640590789023705026710858, −8.297706315180732564783530237341, −7.61570405210692853702276394391, −7.56868881947706170383289424610, −6.98691450681450810623797432226, −6.95226098665555802628666913861, −6.20664014723661831520743104943, −5.70647041365490591993336614399, −5.42452230654057861368683462675, −5.15532546922596569532937510490, −4.31318467793165871745320902589, −3.84786565396883117942173891020, −3.62040442589733917764901550062, −3.60392226358927807982553663471, −2.32553874092456749652720793604, −1.96697823764337574941749382264, −1.13245979957493476425247304735, −0.823615537290723055001011782252,
0.823615537290723055001011782252, 1.13245979957493476425247304735, 1.96697823764337574941749382264, 2.32553874092456749652720793604, 3.60392226358927807982553663471, 3.62040442589733917764901550062, 3.84786565396883117942173891020, 4.31318467793165871745320902589, 5.15532546922596569532937510490, 5.42452230654057861368683462675, 5.70647041365490591993336614399, 6.20664014723661831520743104943, 6.95226098665555802628666913861, 6.98691450681450810623797432226, 7.56868881947706170383289424610, 7.61570405210692853702276394391, 8.297706315180732564783530237341, 8.964739640590789023705026710858, 9.162980863271723136730212827289, 9.353836511598731687283558502368
