| L(s) = 1 | + 4·4-s + 8·7-s + 24·11-s + 12·16-s − 24·23-s + 64·25-s + 32·28-s − 120·29-s − 80·37-s − 128·43-s + 96·44-s + 22·49-s + 216·53-s + 32·64-s + 176·67-s + 120·71-s + 192·77-s + 128·79-s − 96·92-s + 256·100-s + 168·107-s − 8·109-s + 96·112-s + 360·113-s − 480·116-s − 88·121-s + 127-s + ⋯ |
| L(s) = 1 | + 4-s + 8/7·7-s + 2.18·11-s + 3/4·16-s − 1.04·23-s + 2.55·25-s + 8/7·28-s − 4.13·29-s − 2.16·37-s − 2.97·43-s + 2.18·44-s + 0.448·49-s + 4.07·53-s + 1/2·64-s + 2.62·67-s + 1.69·71-s + 2.49·77-s + 1.62·79-s − 1.04·92-s + 2.55·100-s + 1.57·107-s − 0.0733·109-s + 6/7·112-s + 3.18·113-s − 4.13·116-s − 0.727·121-s + 0.00787·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.210841986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.210841986\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 8 T + 6 p T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 64 T^{2} + 1986 T^{4} - 64 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 12 T + 260 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 244 T^{2} + 53574 T^{4} - 244 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $C_2^3$ | \( 1 + 320 T^{2} + 546 p^{2} T^{4} + 320 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 652 T^{2} + 348486 T^{4} - 652 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 12 T + 932 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{4} \) |
| 31 | $D_4\times C_2$ | \( 1 - 1252 T^{2} + 745926 T^{4} - 1252 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 40 T + 546 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 4960 T^{2} + 11110434 T^{4} - 4960 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 64 T + 2922 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 3940 T^{2} + 5766 p^{2} T^{4} - 3940 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 108 T + 8246 T^{2} - 108 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 4420 T^{2} + 6539622 T^{4} - 4420 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 14596 T^{2} + 80934054 T^{4} - 14596 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 88 T + 10626 T^{2} - 88 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 60 T + 4484 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 13108 T^{2} + 98972646 T^{4} - 13108 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 64 T + 3138 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 16612 T^{2} + 134415078 T^{4} - 16612 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 8896 T^{2} + 52680354 T^{4} - 8896 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 24820 T^{2} + 317127462 T^{4} - 24820 p^{4} T^{6} + p^{8} T^{8} \) |
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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
−9.677626717277536076238031986941, −9.095807131029857804267062996500, −9.018985500318887422018445794702, −8.859971293110088272252780814349, −8.496336899267924484592007137846, −8.229454515596870104978422256300, −7.903612847393583452137338853795, −7.43083754234991012902611636781, −7.28407555501163355741561862090, −6.81603207757209924377277095503, −6.73887302093743430002416618475, −6.67375099281273375981940689910, −5.91250392134167883564870894621, −5.83784777941426809633355268342, −5.15380047341106742011838771058, −5.13515569714500388776960448585, −4.90243776562815311854785785491, −3.86027714167018860061238907070, −3.81604870324180334936480392756, −3.77263563162550108011292070486, −3.16177521355316739651267637864, −2.24902481766302355268242120835, −1.92308933130853592037675298565, −1.67046277916240950317221356441, −0.862224439222901504643703981763,
0.862224439222901504643703981763, 1.67046277916240950317221356441, 1.92308933130853592037675298565, 2.24902481766302355268242120835, 3.16177521355316739651267637864, 3.77263563162550108011292070486, 3.81604870324180334936480392756, 3.86027714167018860061238907070, 4.90243776562815311854785785491, 5.13515569714500388776960448585, 5.15380047341106742011838771058, 5.83784777941426809633355268342, 5.91250392134167883564870894621, 6.67375099281273375981940689910, 6.73887302093743430002416618475, 6.81603207757209924377277095503, 7.28407555501163355741561862090, 7.43083754234991012902611636781, 7.903612847393583452137338853795, 8.229454515596870104978422256300, 8.496336899267924484592007137846, 8.859971293110088272252780814349, 9.018985500318887422018445794702, 9.095807131029857804267062996500, 9.677626717277536076238031986941
