| L(s) = 1 | − 5-s − 9-s + 13·13-s − 6·17-s − 9·25-s − 9·29-s − 4·37-s − 12·41-s + 45-s + 9·49-s + 9·53-s + 7·61-s − 13·65-s + 14·73-s + 81-s + 6·85-s + 6·89-s − 22·97-s − 11·101-s − 19·109-s + 14·113-s − 13·117-s + 4·121-s + 14·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1/3·9-s + 3.60·13-s − 1.45·17-s − 9/5·25-s − 1.67·29-s − 0.657·37-s − 1.87·41-s + 0.149·45-s + 9/7·49-s + 1.23·53-s + 0.896·61-s − 1.61·65-s + 1.63·73-s + 1/9·81-s + 0.650·85-s + 0.635·89-s − 2.23·97-s − 1.09·101-s − 1.81·109-s + 1.31·113-s − 1.20·117-s + 4/11·121-s + 1.25·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 17 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 103 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−7.994378785909277882163835328488, −7.16952830993186147034698249603, −6.92182307757914964067059689545, −6.40125764094875033342219848712, −6.00470476794720355886271098191, −5.61018623635099960531852211574, −5.30453850497600830393237802572, −4.36552662051385513896048948422, −3.93675475487817579984662759904, −3.65583946789049589307829418786, −3.40390264094066771889683624543, −2.31868631037184266099357886977, −1.82681289124888437315608128205, −1.12443141322978811367166061171, 0,
1.12443141322978811367166061171, 1.82681289124888437315608128205, 2.31868631037184266099357886977, 3.40390264094066771889683624543, 3.65583946789049589307829418786, 3.93675475487817579984662759904, 4.36552662051385513896048948422, 5.30453850497600830393237802572, 5.61018623635099960531852211574, 6.00470476794720355886271098191, 6.40125764094875033342219848712, 6.92182307757914964067059689545, 7.16952830993186147034698249603, 7.994378785909277882163835328488
