| L(s) = 1 | + 2·3-s + 9-s − 6·11-s + 6·23-s − 5·25-s − 4·27-s + 12·31-s − 12·33-s + 14·37-s − 10·47-s − 4·49-s + 2·53-s + 6·67-s + 12·69-s − 8·71-s − 10·75-s − 11·81-s − 12·89-s + 24·93-s + 18·97-s − 6·99-s + 2·103-s + 28·111-s − 10·113-s + 25·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s − 1.80·11-s + 1.25·23-s − 25-s − 0.769·27-s + 2.15·31-s − 2.08·33-s + 2.30·37-s − 1.45·47-s − 4/7·49-s + 0.274·53-s + 0.733·67-s + 1.44·69-s − 0.949·71-s − 1.15·75-s − 1.22·81-s − 1.27·89-s + 2.48·93-s + 1.82·97-s − 0.603·99-s + 0.197·103-s + 2.65·111-s − 0.940·113-s + 2.27·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.521578241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.521578241\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 112 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
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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
−7.907444021284946561777834717018, −7.67510306157285013697580309036, −7.08532848854378419687884107365, −6.62553837034146544342954668647, −6.05317531474568250854484221588, −5.71017817439842274703684319549, −5.16290119808022107284196815870, −4.59973172753640026312431303249, −4.41323085135148745141414380148, −3.57239682091301333666568069668, −3.10178535386571411619341760327, −2.65678826184174185856159063562, −2.42816523499942208017604098067, −1.59587711328906638623066423622, −0.60370707306457278843549771030,
0.60370707306457278843549771030, 1.59587711328906638623066423622, 2.42816523499942208017604098067, 2.65678826184174185856159063562, 3.10178535386571411619341760327, 3.57239682091301333666568069668, 4.41323085135148745141414380148, 4.59973172753640026312431303249, 5.16290119808022107284196815870, 5.71017817439842274703684319549, 6.05317531474568250854484221588, 6.62553837034146544342954668647, 7.08532848854378419687884107365, 7.67510306157285013697580309036, 7.907444021284946561777834717018
