| 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
| \(L(1)\) |
\(\approx\) |
\(1.368055916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.368055916\) |
| \(L(\frac{3}{2})\) |
|
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
−7.57992738761191657202383396808, −7.01588506882836258599405623548, −6.84775663259973337774910849398, −6.83332850925299462284188648999, −6.71368811642471717302431040962, −6.21752813489160752839231896883, −6.05194532522764385833944496954, −5.66507361989607478536249095400, −5.62772530751221583752710294768, −5.17121339150905459758234223314, −4.91999316105519288251409516356, −4.82030855904975558518792360855, −4.78066862679563805895927793679, −4.13399543037172595332335176138, −4.07998570040077863170079467191, −3.88490084153041906223180962243, −3.37140658726089546864194931169, −3.11232730619813553288894916702, −3.03795318017681575240638062702, −2.62124695530669829008843234732, −2.33752077307676352294556198400, −1.57529871934387436279978745277, −1.02081493097523189981406554888, −0.981546609604123035453254277812, −0.47925865927692612212045167709,
0.47925865927692612212045167709, 0.981546609604123035453254277812, 1.02081493097523189981406554888, 1.57529871934387436279978745277, 2.33752077307676352294556198400, 2.62124695530669829008843234732, 3.03795318017681575240638062702, 3.11232730619813553288894916702, 3.37140658726089546864194931169, 3.88490084153041906223180962243, 4.07998570040077863170079467191, 4.13399543037172595332335176138, 4.78066862679563805895927793679, 4.82030855904975558518792360855, 4.91999316105519288251409516356, 5.17121339150905459758234223314, 5.62772530751221583752710294768, 5.66507361989607478536249095400, 6.05194532522764385833944496954, 6.21752813489160752839231896883, 6.71368811642471717302431040962, 6.83332850925299462284188648999, 6.84775663259973337774910849398, 7.01588506882836258599405623548, 7.57992738761191657202383396808
