| L(s) = 1 | + 2·3-s − 2·5-s + 2·9-s + 4·11-s + 12·13-s − 4·15-s − 2·19-s − 8·23-s + 6·25-s − 8·27-s − 4·31-s + 8·33-s + 24·39-s − 16·41-s + 8·43-s − 4·45-s − 12·47-s + 20·53-s − 8·55-s − 4·57-s + 14·59-s − 18·61-s − 24·65-s − 8·67-s − 16·69-s + 16·71-s − 12·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 2/3·9-s + 1.20·11-s + 3.32·13-s − 1.03·15-s − 0.458·19-s − 1.66·23-s + 6/5·25-s − 1.53·27-s − 0.718·31-s + 1.39·33-s + 3.84·39-s − 2.49·41-s + 1.21·43-s − 0.596·45-s − 1.75·47-s + 2.74·53-s − 1.07·55-s − 0.529·57-s + 1.82·59-s − 2.30·61-s − 2.97·65-s − 0.977·67-s − 1.92·69-s + 1.89·71-s − 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.221579890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.221579890\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 - 2 T + 2 T^{2} + 8 T^{3} - 17 T^{4} + 8 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 2 T - 2 T^{2} - 8 T^{3} - 9 T^{4} - 8 p T^{5} - 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_4\times C_2$ | \( 1 - 4 T + 10 T^{2} + 64 T^{3} - 261 T^{4} + 64 p T^{5} + 10 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $D_4\times C_2$ | \( 1 + 2 T - 30 T^{2} - 8 T^{3} + 719 T^{4} - 8 p T^{5} - 30 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_4\times C_2$ | \( 1 + 4 T - 30 T^{2} - 64 T^{3} + 659 T^{4} - 64 p T^{5} - 30 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 - 54 T^{2} + 1547 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 12 T + 34 T^{2} + 192 T^{3} + 3123 T^{4} + 192 p T^{5} + 34 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 14 T + 34 T^{2} - 616 T^{3} + 11199 T^{4} - 616 p T^{5} + 34 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 18 T + 126 T^{2} + 1368 T^{3} + 15719 T^{4} + 1368 p T^{5} + 126 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 12 T + 42 T^{2} - 528 T^{3} - 5437 T^{4} - 528 p T^{5} + 42 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 8 T - 30 T^{2} - 512 T^{3} - 2461 T^{4} - 512 p T^{5} - 30 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 14 T + 210 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.73572501906924124958186749676, −6.58073372645822217537051123570, −6.42761628732282576191671602002, −6.01522672751895765906923449138, −5.86523092570553318608334144575, −5.77691905411610541340483041319, −5.55234315520810434367549661619, −5.21009671979036581161256219487, −4.94107167724214805076295959782, −4.47663708869368072595324037714, −4.46111152451508839729537067256, −4.08446188428997963549101603103, −3.79698702164160326785658745577, −3.71876041416863826967944527129, −3.70767041938963134755585023111, −3.56913284623922601239430305720, −3.08409200726652853137110515196, −2.78338691584929939803169486455, −2.64483622295355450505381806943, −2.06329410952460123809228228887, −1.68243890793245558774064155403, −1.64672637503760675154624235840, −1.44409773218803673747352396315, −0.867939543466546472513418044335, −0.36759290674107369404878430666,
0.36759290674107369404878430666, 0.867939543466546472513418044335, 1.44409773218803673747352396315, 1.64672637503760675154624235840, 1.68243890793245558774064155403, 2.06329410952460123809228228887, 2.64483622295355450505381806943, 2.78338691584929939803169486455, 3.08409200726652853137110515196, 3.56913284623922601239430305720, 3.70767041938963134755585023111, 3.71876041416863826967944527129, 3.79698702164160326785658745577, 4.08446188428997963549101603103, 4.46111152451508839729537067256, 4.47663708869368072595324037714, 4.94107167724214805076295959782, 5.21009671979036581161256219487, 5.55234315520810434367549661619, 5.77691905411610541340483041319, 5.86523092570553318608334144575, 6.01522672751895765906923449138, 6.42761628732282576191671602002, 6.58073372645822217537051123570, 6.73572501906924124958186749676
