| L(s) = 1 | − 3·3-s − 6·5-s − 8·7-s + 2·9-s − 6·11-s − 6·13-s + 18·15-s − 9·17-s + 24·21-s + 17·25-s + 3·27-s + 12·29-s + 3·31-s + 18·33-s + 48·35-s − 30·37-s + 18·39-s + 16·43-s − 12·45-s − 18·47-s + 34·49-s + 27·51-s − 12·53-s + 36·55-s + 15·59-s − 21·61-s − 16·63-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 2.68·5-s − 3.02·7-s + 2/3·9-s − 1.80·11-s − 1.66·13-s + 4.64·15-s − 2.18·17-s + 5.23·21-s + 17/5·25-s + 0.577·27-s + 2.22·29-s + 0.538·31-s + 3.13·33-s + 8.11·35-s − 4.93·37-s + 2.88·39-s + 2.43·43-s − 1.78·45-s − 2.62·47-s + 34/7·49-s + 3.78·51-s − 1.64·53-s + 4.85·55-s + 1.95·59-s − 2.68·61-s − 2.01·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + p T^{2} )^{2}( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 9 T + 35 T^{2} + 108 T^{3} + 450 T^{4} + 108 p T^{5} + 35 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 - 5 T^{2} - 336 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 13 T^{2} - 360 T^{4} - 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 3 T + 19 T^{2} + 216 T^{3} - 1140 T^{4} + 216 p T^{5} + 19 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 15 T + 112 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 145 T^{2} + 8544 T^{4} - 145 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 + 18 T + 185 T^{2} + 1386 T^{3} + 8796 T^{4} + 1386 p T^{5} + 185 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 12 T + 67 T^{2} + 228 T^{3} + 96 T^{4} + 228 p T^{5} + 67 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 - 15 T + p T^{2} )^{2}( 1 + 15 T + 166 T^{2} + 15 p T^{3} + p^{2} T^{4} ) \) |
| 61 | $D_4\times C_2$ | \( 1 + 21 T + 281 T^{2} + 2814 T^{3} + 23202 T^{4} + 2814 p T^{5} + 281 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + T - 59 T^{2} - 74 T^{3} - 956 T^{4} - 74 p T^{5} - 59 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 15 T + 31 T^{2} - 720 T^{3} + 15882 T^{4} - 720 p T^{5} + 31 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 15 T + 227 T^{2} + 2280 T^{3} + 22788 T^{4} + 2280 p T^{5} + 227 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 256 T^{2} + 28974 T^{4} - 256 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 9 T + 209 T^{2} + 1638 T^{3} + 27606 T^{4} + 1638 p T^{5} + 209 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
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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
−8.263207969780276657435456624370, −7.78184817041447703935316236332, −7.76942499375521531478034083886, −7.50833656425156768445201519381, −7.11831484963757485361880833535, −7.01996070435458445609873312599, −6.70548167426787555578508219730, −6.62435172524515034387642958077, −6.52223587519607674426369943540, −6.06186017661489557523087551785, −6.00128705372451731846737384341, −5.48235989149859266640484355399, −5.26764250985512877787777979164, −4.90013406336915172262242600107, −4.89655158930108256851066881937, −4.63640974008494304751027397792, −4.28152165387184610930481707235, −3.93467374713835718703921688591, −3.57693955005626239367065238214, −3.33143979393571803551609763750, −3.21115798395136815651509541079, −2.69865561473621833195540285660, −2.53890952103811027235224814577, −2.28789419963513714167376894489, −1.20070044324195019316037437008, 0, 0, 0, 0,
1.20070044324195019316037437008, 2.28789419963513714167376894489, 2.53890952103811027235224814577, 2.69865561473621833195540285660, 3.21115798395136815651509541079, 3.33143979393571803551609763750, 3.57693955005626239367065238214, 3.93467374713835718703921688591, 4.28152165387184610930481707235, 4.63640974008494304751027397792, 4.89655158930108256851066881937, 4.90013406336915172262242600107, 5.26764250985512877787777979164, 5.48235989149859266640484355399, 6.00128705372451731846737384341, 6.06186017661489557523087551785, 6.52223587519607674426369943540, 6.62435172524515034387642958077, 6.70548167426787555578508219730, 7.01996070435458445609873312599, 7.11831484963757485361880833535, 7.50833656425156768445201519381, 7.76942499375521531478034083886, 7.78184817041447703935316236332, 8.263207969780276657435456624370
