| L(s) = 1 | − 4·3-s − 4·4-s + 10·9-s + 16·12-s + 16·13-s + 7·16-s + 16·17-s − 20·27-s + 8·29-s − 40·36-s − 64·39-s + 24·47-s − 28·48-s − 64·51-s − 64·52-s − 8·64-s − 64·68-s + 16·71-s + 16·73-s − 32·79-s + 35·81-s + 24·83-s − 32·87-s + 48·97-s + 80·108-s + 8·109-s − 32·116-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 2·4-s + 10/3·9-s + 4.61·12-s + 4.43·13-s + 7/4·16-s + 3.88·17-s − 3.84·27-s + 1.48·29-s − 6.66·36-s − 10.2·39-s + 3.50·47-s − 4.04·48-s − 8.96·51-s − 8.87·52-s − 64-s − 7.76·68-s + 1.89·71-s + 1.87·73-s − 3.60·79-s + 35/9·81-s + 2.63·83-s − 3.43·87-s + 4.87·97-s + 7.69·108-s + 0.766·109-s − 2.97·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.919565099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.919565099\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{4} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p^{2} T^{2} + 9 T^{4} + p^{4} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 19 | $D_4\times C_2$ | \( 1 + 48 T^{2} + 1106 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 40 T^{2} + 1266 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 36 T^{2} - 10 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 124 T^{2} + 7734 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 160 T^{2} + 12114 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 60 T^{2} + 6806 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 8 T + 114 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 16 T + 210 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 89 | $D_4\times C_2$ | \( 1 + 100 T^{2} + 6054 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 24 T + 290 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.02916235373375602408616444114, −5.71372743536268795241716935557, −5.57136042257958697844227705808, −5.53641517608998441110701142752, −5.51998244030968874708798296047, −4.99237937445917782148512611504, −4.73493628508467850837772194489, −4.71463410664560802802468520704, −4.70414068967458003378249543241, −4.07572911158261936234386986123, −3.93611030162620368147921238470, −3.88506587307919313930732029910, −3.82553557400490251686701075571, −3.45882197710195971320529396808, −3.25377328401529988209346447505, −3.20201120635432632656029406049, −2.75034245525125024705289118519, −2.36238977081349655492779706471, −1.82008814499021281810217543221, −1.51266737127042097706832080440, −1.47377976052566319837718061936, −0.854637472226835135908143334879, −0.837759581689706862809287524556, −0.71414800560599163514864004153, −0.66289297090853677245546156900,
0.66289297090853677245546156900, 0.71414800560599163514864004153, 0.837759581689706862809287524556, 0.854637472226835135908143334879, 1.47377976052566319837718061936, 1.51266737127042097706832080440, 1.82008814499021281810217543221, 2.36238977081349655492779706471, 2.75034245525125024705289118519, 3.20201120635432632656029406049, 3.25377328401529988209346447505, 3.45882197710195971320529396808, 3.82553557400490251686701075571, 3.88506587307919313930732029910, 3.93611030162620368147921238470, 4.07572911158261936234386986123, 4.70414068967458003378249543241, 4.71463410664560802802468520704, 4.73493628508467850837772194489, 4.99237937445917782148512611504, 5.51998244030968874708798296047, 5.53641517608998441110701142752, 5.57136042257958697844227705808, 5.71372743536268795241716935557, 6.02916235373375602408616444114
