| L(s) = 1 | + 4·13-s − 6·17-s − 12·23-s + 25-s + 2·43-s + 5·49-s + 12·53-s − 16·61-s − 20·79-s − 24·101-s − 28·103-s − 24·107-s + 12·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 1.10·13-s − 1.45·17-s − 2.50·23-s + 1/5·25-s + 0.304·43-s + 5/7·49-s + 1.64·53-s − 2.04·61-s − 2.25·79-s − 2.38·101-s − 2.75·103-s − 2.32·107-s + 1.12·113-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.320025303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.320025303\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 83 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−9.812461822628167761622544078993, −8.915429172866677088206285614105, −8.606478365697303614063029572112, −8.424957777006314628501732890005, −7.81138889037234542002277677834, −7.58896288325398766625253121248, −7.02010058723721281695404094607, −6.52106398146623337112226896597, −6.36809425848024170526499657408, −5.71009037203431896829016647790, −5.64443880525340334712489425347, −4.96758791795326287802478599269, −4.16905535392628343059553087980, −4.14462886024278059553455071502, −3.86190333066788868599621640257, −2.85532633718420664704111697882, −2.67951995157547332753661561742, −1.82195188073933874151908719044, −1.52054806932982127235839306587, −0.41450867903394054016512981549,
0.41450867903394054016512981549, 1.52054806932982127235839306587, 1.82195188073933874151908719044, 2.67951995157547332753661561742, 2.85532633718420664704111697882, 3.86190333066788868599621640257, 4.14462886024278059553455071502, 4.16905535392628343059553087980, 4.96758791795326287802478599269, 5.64443880525340334712489425347, 5.71009037203431896829016647790, 6.36809425848024170526499657408, 6.52106398146623337112226896597, 7.02010058723721281695404094607, 7.58896288325398766625253121248, 7.81138889037234542002277677834, 8.424957777006314628501732890005, 8.606478365697303614063029572112, 8.915429172866677088206285614105, 9.812461822628167761622544078993
