| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 4·13-s + 5·16-s + 6·17-s + 4·19-s + 8·25-s − 8·26-s − 6·32-s − 12·34-s − 8·38-s − 8·43-s + 24·47-s − 4·49-s − 16·50-s + 12·52-s − 12·53-s − 12·59-s + 7·64-s − 20·67-s + 18·68-s + 12·76-s + 12·83-s + 16·86-s − 12·89-s − 48·94-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s + 1.10·13-s + 5/4·16-s + 1.45·17-s + 0.917·19-s + 8/5·25-s − 1.56·26-s − 1.06·32-s − 2.05·34-s − 1.29·38-s − 1.21·43-s + 3.50·47-s − 4/7·49-s − 2.26·50-s + 1.66·52-s − 1.64·53-s − 1.56·59-s + 7/8·64-s − 2.44·67-s + 2.18·68-s + 1.37·76-s + 1.31·83-s + 1.72·86-s − 1.27·89-s − 4.95·94-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93636 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93636 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9071732046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9071732046\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 17 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
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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
−12.02620659240219112526598306365, −11.13230241228995560910523046248, −10.88063574061265581732493567012, −10.63551944026625307654854089393, −9.990874840833729381787997709838, −9.527277597510449659511349470592, −9.178900121570348977684715890071, −8.651352448882167569363164085531, −8.198354951513115824164346223962, −7.77879977604317506683805015754, −7.17694207720659108003448634618, −6.88595554935714671088420439183, −5.98832854861153248731297336061, −5.81211010404588205818654087297, −5.02455124867949836283567224813, −4.17281046156135786819454367902, −3.11140308465618835864316615034, −3.04551675307913225849244835951, −1.63882728095410674323939409728, −0.987150374847621157800648176480,
0.987150374847621157800648176480, 1.63882728095410674323939409728, 3.04551675307913225849244835951, 3.11140308465618835864316615034, 4.17281046156135786819454367902, 5.02455124867949836283567224813, 5.81211010404588205818654087297, 5.98832854861153248731297336061, 6.88595554935714671088420439183, 7.17694207720659108003448634618, 7.77879977604317506683805015754, 8.198354951513115824164346223962, 8.651352448882167569363164085531, 9.178900121570348977684715890071, 9.527277597510449659511349470592, 9.990874840833729381787997709838, 10.63551944026625307654854089393, 10.88063574061265581732493567012, 11.13230241228995560910523046248, 12.02620659240219112526598306365
