| L(s) = 1 | − 2·3-s + 4·5-s + 2·9-s − 8·15-s + 12·19-s − 8·23-s − 6·27-s − 28·29-s + 20·43-s + 8·45-s + 32·47-s + 16·49-s + 4·53-s − 24·57-s + 12·67-s + 16·69-s − 8·71-s + 8·73-s + 11·81-s + 56·87-s + 48·95-s − 16·97-s + 20·101-s − 32·115-s + 16·121-s − 20·125-s + 127-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s + 2/3·9-s − 2.06·15-s + 2.75·19-s − 1.66·23-s − 1.15·27-s − 5.19·29-s + 3.04·43-s + 1.19·45-s + 4.66·47-s + 16/7·49-s + 0.549·53-s − 3.17·57-s + 1.46·67-s + 1.92·69-s − 0.949·71-s + 0.936·73-s + 11/9·81-s + 6.00·87-s + 4.92·95-s − 1.62·97-s + 1.99·101-s − 2.98·115-s + 1.45·121-s − 1.78·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.488575444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.488575444\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $D_{4}$ | \( ( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 142 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_4\times C_2$ | \( 1 - 16 T^{2} + 126 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 358 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 6 T + 42 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 14 T + 102 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 96 T^{2} + 4046 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 2902 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6406 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 10 T + 106 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 53 | $D_{4}$ | \( ( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 224 T^{2} + 19486 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 172 T^{2} + 13558 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T - 34 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 79 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 7758 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 272 T^{2} + 31774 T^{4} - 272 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 162 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.44247778666628814458508171474, −7.19509138506185627797900867530, −7.05093251566331885525166915368, −6.78943000404672201453841396795, −6.27909751126357004330830435575, −5.97262787248073421141463675430, −5.80571108087965117853067408926, −5.70312602719558261373858841543, −5.60956950257273861125774565844, −5.54760393601450449507557420544, −5.40736040874143492967607809426, −5.08875230228693152536572343495, −4.30563125799170975452305091893, −4.21776008711077675115181548693, −4.13779310981442402282961542186, −3.81326120014705840383825473100, −3.42331505601323547909248293365, −3.36482183168630725341692540379, −2.54211825191188834926676242480, −2.29049025474332987966901383720, −2.17739300799737743999073316964, −1.96498566430316388864557059423, −1.43455523546007024285684201262, −0.925651741610071157297040918962, −0.51125426464725027367664674257,
0.51125426464725027367664674257, 0.925651741610071157297040918962, 1.43455523546007024285684201262, 1.96498566430316388864557059423, 2.17739300799737743999073316964, 2.29049025474332987966901383720, 2.54211825191188834926676242480, 3.36482183168630725341692540379, 3.42331505601323547909248293365, 3.81326120014705840383825473100, 4.13779310981442402282961542186, 4.21776008711077675115181548693, 4.30563125799170975452305091893, 5.08875230228693152536572343495, 5.40736040874143492967607809426, 5.54760393601450449507557420544, 5.60956950257273861125774565844, 5.70312602719558261373858841543, 5.80571108087965117853067408926, 5.97262787248073421141463675430, 6.27909751126357004330830435575, 6.78943000404672201453841396795, 7.05093251566331885525166915368, 7.19509138506185627797900867530, 7.44247778666628814458508171474
