| L(s) = 1 | − 8·7-s − 32·13-s + 32·19-s + 8·25-s − 8·31-s − 104·37-s + 112·43-s − 156·49-s − 120·61-s + 336·67-s − 112·73-s − 152·79-s + 256·91-s − 352·97-s − 344·103-s − 512·109-s − 12·121-s + 127-s + 131-s − 256·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 8/7·7-s − 2.46·13-s + 1.68·19-s + 8/25·25-s − 0.258·31-s − 2.81·37-s + 2.60·43-s − 3.18·49-s − 1.96·61-s + 5.01·67-s − 1.53·73-s − 1.92·79-s + 2.81·91-s − 3.62·97-s − 3.33·103-s − 4.69·109-s − 0.0991·121-s + 0.00787·127-s + 0.00763·131-s − 1.92·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5662158972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5662158972\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 914 T^{4} - 8 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{4} \) |
| 11 | $D_4\times C_2$ | \( 1 + 12 T^{2} - 21370 T^{4} + 12 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 16 T + 314 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 800 T^{2} + 325634 T^{4} - 800 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 16 T + 434 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 564 T^{2} + 436454 T^{4} - 564 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2600 T^{2} + 3045074 T^{4} - 2600 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 518 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 5888 T^{2} + 14148290 T^{4} - 5888 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 56 T + 3074 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 1652 T^{2} + 2309030 T^{4} - 1652 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 8680 T^{2} + 33219474 T^{4} - 8680 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 1554 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 60 T + 6934 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 168 T + 15682 T^{2} - 168 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12596 T^{2} + 79584614 T^{4} - 12596 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 56 T - 1230 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 76 T + 12518 T^{2} + 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 24244 T^{2} + 241403334 T^{4} - 24244 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 10176 T^{2} + 39402434 T^{4} - 10176 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 176 T + 26210 T^{2} + 176 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.87980294258448752850096858923, −6.70229029296151421795473810774, −6.47952713502294414229071366743, −6.18753681458099974202219078113, −5.84471345802896402781927872816, −5.57197511742230443302142779665, −5.48410867334203188656611632531, −5.19800149977966282920248298780, −4.97917310888126054859605780486, −4.80881362519043683380618168546, −4.78640050671682994586835282706, −4.02154432303256850825136946439, −3.98045599237444633353546059917, −3.85707025730444270589982389739, −3.56786238889913830831702267602, −3.02231610544417963919742775188, −2.86625530705419962280918156751, −2.73857402555764682105509436760, −2.64336087446471826629011314015, −2.17144845055516357497557692129, −1.53355656773322042435039979355, −1.52987413152206895160893167529, −1.18364812921292370210566993123, −0.36263584574667939398476623649, −0.19668035345980622432827921601,
0.19668035345980622432827921601, 0.36263584574667939398476623649, 1.18364812921292370210566993123, 1.52987413152206895160893167529, 1.53355656773322042435039979355, 2.17144845055516357497557692129, 2.64336087446471826629011314015, 2.73857402555764682105509436760, 2.86625530705419962280918156751, 3.02231610544417963919742775188, 3.56786238889913830831702267602, 3.85707025730444270589982389739, 3.98045599237444633353546059917, 4.02154432303256850825136946439, 4.78640050671682994586835282706, 4.80881362519043683380618168546, 4.97917310888126054859605780486, 5.19800149977966282920248298780, 5.48410867334203188656611632531, 5.57197511742230443302142779665, 5.84471345802896402781927872816, 6.18753681458099974202219078113, 6.47952713502294414229071366743, 6.70229029296151421795473810774, 6.87980294258448752850096858923
