| L(s) = 1 | + 2·2-s + 2·4-s − 4·5-s + 4·8-s − 8·10-s − 6·13-s + 8·16-s − 4·17-s − 8·20-s + 5·25-s − 12·26-s − 30·29-s + 8·32-s − 8·34-s − 22·37-s − 16·40-s + 24·41-s + 10·50-s − 12·52-s − 10·53-s − 60·58-s − 12·61-s + 8·64-s + 24·65-s − 8·68-s + 22·73-s − 44·74-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 1.78·5-s + 1.41·8-s − 2.52·10-s − 1.66·13-s + 2·16-s − 0.970·17-s − 1.78·20-s + 25-s − 2.35·26-s − 5.57·29-s + 1.41·32-s − 1.37·34-s − 3.61·37-s − 2.52·40-s + 3.74·41-s + 1.41·50-s − 1.66·52-s − 1.37·53-s − 7.87·58-s − 1.53·61-s + 64-s + 2.97·65-s − 0.970·68-s + 2.57·73-s − 5.11·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.300859861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.300859861\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| good | 3 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 7 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + 10 T + p T^{2} )^{2}( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + 12 T + p T^{2} )^{2}( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} ) \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} ) \) |
| 43 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 59 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 + 12 T + p T^{2} )^{2}( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} ) \) |
| 67 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} )( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 - 8 T - 33 T^{2} - 8 p T^{3} + p^{2} T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.792035243466649634966189173722, −8.400247076931231053490619797675, −7.87783439236222089542648807109, −7.79550374872632366999366220395, −7.60878650336458469676493595936, −7.55302108494919958138867422268, −7.11521259470731472799974629183, −7.01104468523068032294358559032, −6.91142190369548993287531999010, −6.12655603399409297033607756076, −5.89933104727720344581735926307, −5.67732661208303335685770569564, −5.56958272668486288646026675146, −4.86561018704451972422094923930, −4.80979600130612078434710185279, −4.49988109252360874053445407454, −4.49210578042835974567409388455, −3.76036702768594012231891319056, −3.53397698297726739596700015736, −3.46000644685835524707585614927, −3.37800539749564835381506697996, −2.13845651884361027847611206064, −2.08898932961594158668462306208, −1.95942638812661455191096502919, −0.43759024778853351831377871820,
0.43759024778853351831377871820, 1.95942638812661455191096502919, 2.08898932961594158668462306208, 2.13845651884361027847611206064, 3.37800539749564835381506697996, 3.46000644685835524707585614927, 3.53397698297726739596700015736, 3.76036702768594012231891319056, 4.49210578042835974567409388455, 4.49988109252360874053445407454, 4.80979600130612078434710185279, 4.86561018704451972422094923930, 5.56958272668486288646026675146, 5.67732661208303335685770569564, 5.89933104727720344581735926307, 6.12655603399409297033607756076, 6.91142190369548993287531999010, 7.01104468523068032294358559032, 7.11521259470731472799974629183, 7.55302108494919958138867422268, 7.60878650336458469676493595936, 7.79550374872632366999366220395, 7.87783439236222089542648807109, 8.400247076931231053490619797675, 8.792035243466649634966189173722
