| L(s) = 1 | + 3-s + 4-s + 4·5-s + 9-s + 12-s + 4·15-s − 3·16-s − 4·17-s + 4·20-s + 6·25-s + 27-s + 36-s + 4·37-s − 4·41-s − 8·43-s + 4·45-s + 16·47-s − 3·48-s − 4·51-s + 8·59-s + 4·60-s − 7·64-s − 8·67-s − 4·68-s + 6·75-s − 12·80-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 1.78·5-s + 1/3·9-s + 0.288·12-s + 1.03·15-s − 3/4·16-s − 0.970·17-s + 0.894·20-s + 6/5·25-s + 0.192·27-s + 1/6·36-s + 0.657·37-s − 0.624·41-s − 1.21·43-s + 0.596·45-s + 2.33·47-s − 0.433·48-s − 0.560·51-s + 1.04·59-s + 0.516·60-s − 7/8·64-s − 0.977·67-s − 0.485·68-s + 0.692·75-s − 1.34·80-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64827 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64827 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.548034410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.548034410\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.874946691665207286963349289658, −9.374298000981204301178108560312, −8.960529838835116925605965584332, −8.608395371232830668538320621237, −7.85128193226506182531549877684, −7.22692424043679009953402057414, −6.69349594586877387039942479178, −6.32246164171759109730284094977, −5.74559128296708618051353327185, −5.14469951522295166409941716490, −4.46501658965168276811907614429, −3.69686772352574587275278146114, −2.59577852164829663485966617655, −2.33124636415090208557097734318, −1.54476873036536696221011567489,
1.54476873036536696221011567489, 2.33124636415090208557097734318, 2.59577852164829663485966617655, 3.69686772352574587275278146114, 4.46501658965168276811907614429, 5.14469951522295166409941716490, 5.74559128296708618051353327185, 6.32246164171759109730284094977, 6.69349594586877387039942479178, 7.22692424043679009953402057414, 7.85128193226506182531549877684, 8.608395371232830668538320621237, 8.960529838835116925605965584332, 9.374298000981204301178108560312, 9.874946691665207286963349289658
