| L(s) = 1 | + 6·5-s + 2·9-s − 2·13-s + 10·17-s + 18·25-s − 2·29-s − 3·37-s + 12·41-s + 12·45-s − 4·49-s + 20·53-s − 24·61-s − 12·65-s − 28·73-s − 5·81-s + 60·85-s + 2·97-s − 8·101-s + 10·109-s + 6·113-s − 4·117-s − 8·121-s + 30·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 2.68·5-s + 2/3·9-s − 0.554·13-s + 2.42·17-s + 18/5·25-s − 0.371·29-s − 0.493·37-s + 1.87·41-s + 1.78·45-s − 4/7·49-s + 2.74·53-s − 3.07·61-s − 1.48·65-s − 3.27·73-s − 5/9·81-s + 6.50·85-s + 0.203·97-s − 0.796·101-s + 0.957·109-s + 0.564·113-s − 0.369·117-s − 0.727·121-s + 2.68·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.217592442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.217592442\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 60 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.287206703538890802816098620778, −7.63346335043807688618559005501, −7.30755046963916677814942342962, −7.00530443176361758769773822618, −6.16949509912427761558336432798, −5.84847130243960741678488818491, −5.74876062274937447518317916265, −5.26171261471836226720733532289, −4.67245382381093576452409757883, −4.12632200038204767682739962339, −3.24631527298770444288219515422, −2.80981377542535681745065378158, −2.17862415058349637187788752679, −1.56015191346889096338625151082, −1.13475259358610607968488438145,
1.13475259358610607968488438145, 1.56015191346889096338625151082, 2.17862415058349637187788752679, 2.80981377542535681745065378158, 3.24631527298770444288219515422, 4.12632200038204767682739962339, 4.67245382381093576452409757883, 5.26171261471836226720733532289, 5.74876062274937447518317916265, 5.84847130243960741678488818491, 6.16949509912427761558336432798, 7.00530443176361758769773822618, 7.30755046963916677814942342962, 7.63346335043807688618559005501, 8.287206703538890802816098620778
