| L(s) = 1 | − 4·3-s + 4·7-s + 6·9-s − 4·13-s + 4·17-s − 16·21-s + 12·23-s + 4·27-s + 8·29-s + 12·37-s + 16·39-s − 4·43-s + 20·47-s + 8·49-s − 16·51-s − 12·53-s + 16·59-s + 24·61-s + 24·63-s − 12·67-s − 48·69-s − 4·73-s − 37·81-s − 4·83-s − 32·87-s − 40·89-s − 16·91-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 1.51·7-s + 2·9-s − 1.10·13-s + 0.970·17-s − 3.49·21-s + 2.50·23-s + 0.769·27-s + 1.48·29-s + 1.97·37-s + 2.56·39-s − 0.609·43-s + 2.91·47-s + 8/7·49-s − 2.24·51-s − 1.64·53-s + 2.08·59-s + 3.07·61-s + 3.02·63-s − 1.46·67-s − 5.77·69-s − 0.468·73-s − 4.11·81-s − 0.439·83-s − 3.43·87-s − 4.23·89-s − 1.67·91-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
| \(L(1)\) |
\(\approx\) |
\(1.560575168\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.560575168\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 20 T^{3} + 46 T^{4} - 20 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 214 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} - 4 T^{3} - 194 T^{4} - 4 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 12 T^{3} - 178 T^{4} - 12 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1270 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 444 T^{3} + 2542 T^{4} - 444 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 468 T^{3} + 3038 T^{4} - 468 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 164 T^{3} + 3358 T^{4} + 164 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 20 T + 200 T^{2} - 1860 T^{3} + 15182 T^{4} - 1860 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} - 180 T^{3} - 6274 T^{4} - 180 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 13286 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 44 T^{3} - 3602 T^{4} + 44 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 292 T^{2} + 33670 T^{4} - 292 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 196 T^{3} + 3646 T^{4} + 196 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.84791414692095660375457161460, −7.19178016952799043732192889160, −7.02792039720724326908714007697, −7.02092135956106898309799424082, −6.87329739655549275524894629600, −6.36492695149282262106451734685, −6.31975856188716703070499755179, −5.77192655395766669800019985478, −5.76540175381736131906474234092, −5.30494990846781358281979689493, −5.27342294855299720942768869398, −5.24651245417923246231817363517, −4.86474228005612123630928045181, −4.56503433467273257658051402684, −4.28794441549476446965342689571, −4.17894441372168539200704845626, −3.82337829621787622191335772359, −3.15179002666453104968802859659, −2.78991191725055461258056631825, −2.61701189135634492590009735062, −2.56377605174596560865063889237, −1.60688106022990730191255505391, −1.35884744395899685392508024097, −0.78306369536770183707178650247, −0.68495315058353425993022482270,
0.68495315058353425993022482270, 0.78306369536770183707178650247, 1.35884744395899685392508024097, 1.60688106022990730191255505391, 2.56377605174596560865063889237, 2.61701189135634492590009735062, 2.78991191725055461258056631825, 3.15179002666453104968802859659, 3.82337829621787622191335772359, 4.17894441372168539200704845626, 4.28794441549476446965342689571, 4.56503433467273257658051402684, 4.86474228005612123630928045181, 5.24651245417923246231817363517, 5.27342294855299720942768869398, 5.30494990846781358281979689493, 5.76540175381736131906474234092, 5.77192655395766669800019985478, 6.31975856188716703070499755179, 6.36492695149282262106451734685, 6.87329739655549275524894629600, 7.02092135956106898309799424082, 7.02792039720724326908714007697, 7.19178016952799043732192889160, 7.84791414692095660375457161460
