| L(s) = 1 | − 3-s − 3·4-s + 5-s − 2·9-s + 3·12-s − 15-s + 5·16-s − 3·20-s − 12·23-s − 4·25-s + 5·27-s − 31-s + 6·36-s + 6·37-s − 2·45-s + 10·47-s − 5·48-s + 12·49-s − 7·53-s + 12·59-s + 3·60-s − 3·64-s − 5·67-s + 12·69-s − 3·71-s + 4·75-s + 5·80-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 3/2·4-s + 0.447·5-s − 2/3·9-s + 0.866·12-s − 0.258·15-s + 5/4·16-s − 0.670·20-s − 2.50·23-s − 4/5·25-s + 0.962·27-s − 0.179·31-s + 36-s + 0.986·37-s − 0.298·45-s + 1.45·47-s − 0.721·48-s + 12/7·49-s − 0.961·53-s + 1.56·59-s + 0.387·60-s − 3/8·64-s − 0.610·67-s + 1.44·69-s − 0.356·71-s + 0.461·75-s + 0.559·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
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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
−7.41949878048906135622005174579, −6.78412443955357294813716306122, −6.29443683856812784768101205285, −5.87477767941063352231642757437, −5.62105117210868757879626010833, −5.38149760109488770530288296009, −4.75513180799144398252338231050, −4.21793669363561189835315566674, −4.02121820940130964332915653038, −3.60426081327829454842750009307, −2.69384369214365109075554586399, −2.35262794883262996468820257550, −1.55591218994608419117790081314, −0.67572510247103824890075449546, 0,
0.67572510247103824890075449546, 1.55591218994608419117790081314, 2.35262794883262996468820257550, 2.69384369214365109075554586399, 3.60426081327829454842750009307, 4.02121820940130964332915653038, 4.21793669363561189835315566674, 4.75513180799144398252338231050, 5.38149760109488770530288296009, 5.62105117210868757879626010833, 5.87477767941063352231642757437, 6.29443683856812784768101205285, 6.78412443955357294813716306122, 7.41949878048906135622005174579
