| L(s) = 1 | + 4·4-s + 4·16-s − 8·25-s − 4·37-s − 4·43-s − 16·64-s + 44·67-s + 20·79-s − 32·100-s − 4·109-s + 40·121-s + 127-s + 131-s + 137-s + 139-s − 16·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s − 16·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 2·4-s + 16-s − 8/5·25-s − 0.657·37-s − 0.609·43-s − 2·64-s + 5.37·67-s + 2.25·79-s − 3.19·100-s − 0.383·109-s + 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.31·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.153·169-s − 1.21·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.259717519\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.259717519\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 92 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 112 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.032555063951711344137063250386, −7.67560926915879912079886185546, −7.58365130693717008466942491144, −7.12166478652107226530339109384, −7.06990121167278531388781701873, −6.95162811633682740881105368789, −6.58075505144410173824752364124, −6.29147651339654169986309928524, −6.15771216219854371217483104082, −5.96535343522107660029560353359, −5.65366490005888899785754801000, −5.20584833639502952382641569862, −5.02908713655860041858033502440, −4.82912548424657138651501699310, −4.44389664720983271010962041782, −3.95544712116567946792405562081, −3.66194779668353224752148952161, −3.51952462200294649916587556444, −3.25260139742650708884155216828, −2.48901394909076953594804391716, −2.46678554053813862554345726233, −2.20586491300823955334880921721, −1.82784642471843825147680567386, −1.42032966318660902566473438361, −0.59536133950747632232977552167,
0.59536133950747632232977552167, 1.42032966318660902566473438361, 1.82784642471843825147680567386, 2.20586491300823955334880921721, 2.46678554053813862554345726233, 2.48901394909076953594804391716, 3.25260139742650708884155216828, 3.51952462200294649916587556444, 3.66194779668353224752148952161, 3.95544712116567946792405562081, 4.44389664720983271010962041782, 4.82912548424657138651501699310, 5.02908713655860041858033502440, 5.20584833639502952382641569862, 5.65366490005888899785754801000, 5.96535343522107660029560353359, 6.15771216219854371217483104082, 6.29147651339654169986309928524, 6.58075505144410173824752364124, 6.95162811633682740881105368789, 7.06990121167278531388781701873, 7.12166478652107226530339109384, 7.58365130693717008466942491144, 7.67560926915879912079886185546, 8.032555063951711344137063250386
