| L(s) = 1 | + 2·7-s − 6·13-s − 3·19-s + 4·25-s + 10·31-s − 8·37-s − 12·43-s − 10·49-s − 8·61-s + 10·67-s − 20·73-s − 12·91-s − 6·97-s − 4·103-s − 2·109-s + 14·121-s + 127-s + 131-s − 6·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.66·13-s − 0.688·19-s + 4/5·25-s + 1.79·31-s − 1.31·37-s − 1.82·43-s − 1.42·49-s − 1.02·61-s + 1.22·67-s − 2.34·73-s − 1.25·91-s − 0.609·97-s − 0.394·103-s − 0.191·109-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s − 0.520·133-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 + 0.769·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 393984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 393984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.409833161868881999647789046219, −8.069609815519914489320455119876, −7.50025540996041474144448227225, −7.06902747896085448798752727442, −6.60577015772778632308081434571, −6.22338360588089634203656355567, −5.41044638101075521379694279140, −4.93502208768825656096737421951, −4.72521009583633302296313389410, −4.17956214922518689934563526404, −3.26977072003513632149022604130, −2.80404948899121716640718485007, −2.07962069812932255049263590736, −1.39480164573995248134525156384, 0,
1.39480164573995248134525156384, 2.07962069812932255049263590736, 2.80404948899121716640718485007, 3.26977072003513632149022604130, 4.17956214922518689934563526404, 4.72521009583633302296313389410, 4.93502208768825656096737421951, 5.41044638101075521379694279140, 6.22338360588089634203656355567, 6.60577015772778632308081434571, 7.06902747896085448798752727442, 7.50025540996041474144448227225, 8.069609815519914489320455119876, 8.409833161868881999647789046219
