| L(s) = 1 | + 2·3-s + 2·7-s + 2·9-s + 8·13-s − 8·17-s − 8·19-s + 4·21-s + 10·23-s + 6·27-s + 16·39-s − 8·41-s + 14·43-s + 6·47-s + 2·49-s − 16·51-s + 8·53-s − 16·57-s − 8·59-s − 16·61-s + 4·63-s − 6·67-s + 20·69-s + 8·73-s + 16·79-s + 11·81-s − 10·83-s + 16·91-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.755·7-s + 2/3·9-s + 2.21·13-s − 1.94·17-s − 1.83·19-s + 0.872·21-s + 2.08·23-s + 1.15·27-s + 2.56·39-s − 1.24·41-s + 2.13·43-s + 0.875·47-s + 2/7·49-s − 2.24·51-s + 1.09·53-s − 2.11·57-s − 1.04·59-s − 2.04·61-s + 0.503·63-s − 0.733·67-s + 2.40·69-s + 0.936·73-s + 1.80·79-s + 11/9·81-s − 1.09·83-s + 1.67·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.648160288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.648160288\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
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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
−10.55374786445866496768684134962, −10.29097544765293290287876127753, −9.089996388508825785562720946344, −9.052485826671851602418182376027, −8.747004119777754145033551374780, −8.743942219416152535117199098085, −7.83665134145972727486867128390, −7.83028897057092040245201781733, −6.90187234889065872877206094653, −6.68666952001963490752195364968, −6.20712555634431725100270999448, −5.75823904353993206637646944286, −4.78108752864557783268703417048, −4.65076702289541684965087046332, −4.03416747726069029576543962573, −3.59024636686226889727514752069, −2.89515212615180721093441613575, −2.35336550025024280112408643865, −1.75930866726089681030256218148, −0.959948261472095265651885227825,
0.959948261472095265651885227825, 1.75930866726089681030256218148, 2.35336550025024280112408643865, 2.89515212615180721093441613575, 3.59024636686226889727514752069, 4.03416747726069029576543962573, 4.65076702289541684965087046332, 4.78108752864557783268703417048, 5.75823904353993206637646944286, 6.20712555634431725100270999448, 6.68666952001963490752195364968, 6.90187234889065872877206094653, 7.83028897057092040245201781733, 7.83665134145972727486867128390, 8.743942219416152535117199098085, 8.747004119777754145033551374780, 9.052485826671851602418182376027, 9.089996388508825785562720946344, 10.29097544765293290287876127753, 10.55374786445866496768684134962
