| L(s) = 1 | − 2·2-s + 2·4-s − 12·7-s − 4·8-s − 6·9-s + 24·14-s + 8·16-s − 16·17-s + 12·18-s − 4·23-s − 24·28-s − 16·31-s − 8·32-s + 32·34-s − 12·36-s + 8·46-s − 20·47-s + 72·49-s + 48·56-s + 32·62-s + 72·63-s + 8·64-s − 32·68-s + 24·72-s − 24·73-s + 27·81-s − 48·89-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 4.53·7-s − 1.41·8-s − 2·9-s + 6.41·14-s + 2·16-s − 3.88·17-s + 2.82·18-s − 0.834·23-s − 4.53·28-s − 2.87·31-s − 1.41·32-s + 5.48·34-s − 2·36-s + 1.17·46-s − 2.91·47-s + 72/7·49-s + 6.41·56-s + 4.06·62-s + 9.07·63-s + 64-s − 3.88·68-s + 2.82·72-s − 2.80·73-s + 3·81-s − 5.08·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 - 334 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 2702 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 + 1106 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 2578 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 11822 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.239937498225070216462844688811, −8.086455352724038538696910517861, −7.76121909508280913813800886412, −7.40545686288682513669883587165, −7.02413704186125399330087297378, −6.83991228313777432514046497393, −6.66667038379220174521711421834, −6.66113971471140984897574478743, −6.35621550050914965725130044329, −6.09783254104684005017034257290, −6.01243755803567177580865668824, −5.71859329611369152266052286705, −5.37613217394082291767286621006, −5.28825644087335134144193804014, −4.75578264082075523228167653061, −4.03681851143804977452255575565, −4.01110335712876747641387241347, −3.95665241319280244575890575718, −3.31944107786767652339233543733, −3.19189458801171453531223094106, −2.85419609020867071693304274718, −2.69825471313031873362201149768, −2.57993781644300526256428154779, −1.99247995571556711933296400300, −1.50329024133171333961405448836, 0, 0, 0, 0,
1.50329024133171333961405448836, 1.99247995571556711933296400300, 2.57993781644300526256428154779, 2.69825471313031873362201149768, 2.85419609020867071693304274718, 3.19189458801171453531223094106, 3.31944107786767652339233543733, 3.95665241319280244575890575718, 4.01110335712876747641387241347, 4.03681851143804977452255575565, 4.75578264082075523228167653061, 5.28825644087335134144193804014, 5.37613217394082291767286621006, 5.71859329611369152266052286705, 6.01243755803567177580865668824, 6.09783254104684005017034257290, 6.35621550050914965725130044329, 6.66113971471140984897574478743, 6.66667038379220174521711421834, 6.83991228313777432514046497393, 7.02413704186125399330087297378, 7.40545686288682513669883587165, 7.76121909508280913813800886412, 8.086455352724038538696910517861, 8.239937498225070216462844688811
