| 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
| \(L(1)\) |
\(\approx\) |
\(1.752374701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.752374701\) |
| \(L(\frac{3}{2})\) |
|
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 | $D_{4}$ | \( ( 1 + 16 T + 238 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.77871022911992178770189477970, −6.55443480069745489458794101045, −6.21096437715703021896409158587, −6.04699202568584551542536949452, −5.90385733966935066186706146635, −5.63322644018677180133594010184, −5.34091668004399337499352300294, −5.27750310080199303799399355696, −5.00433023383203266845725630757, −4.92744113464667765127445345725, −4.52625896132045319689355619955, −4.21340540097511975254601781891, −4.20086765688749714920559095768, −3.91358325363214017269895142489, −3.81364795434979051880920249280, −3.05334201112883358038135800779, −3.01130244317053824634602620685, −2.47452599232644446512216152154, −2.44068474115014444914559846124, −2.29446371429252660959904335150, −2.14106352375399772421463332286, −1.27834936501295305893782705802, −1.12119898595768259774953155189, −0.881876731799498281997042878081, −0.31518776568869975407091987471,
0.31518776568869975407091987471, 0.881876731799498281997042878081, 1.12119898595768259774953155189, 1.27834936501295305893782705802, 2.14106352375399772421463332286, 2.29446371429252660959904335150, 2.44068474115014444914559846124, 2.47452599232644446512216152154, 3.01130244317053824634602620685, 3.05334201112883358038135800779, 3.81364795434979051880920249280, 3.91358325363214017269895142489, 4.20086765688749714920559095768, 4.21340540097511975254601781891, 4.52625896132045319689355619955, 4.92744113464667765127445345725, 5.00433023383203266845725630757, 5.27750310080199303799399355696, 5.34091668004399337499352300294, 5.63322644018677180133594010184, 5.90385733966935066186706146635, 6.04699202568584551542536949452, 6.21096437715703021896409158587, 6.55443480069745489458794101045, 6.77871022911992178770189477970
