| L(s) = 1 | − 2·3-s − 6·7-s + 5·9-s − 2·11-s + 4·13-s + 6·17-s + 6·19-s + 12·21-s + 6·23-s − 4·25-s − 10·27-s + 6·29-s + 4·33-s + 2·37-s − 8·39-s + 6·41-s − 6·43-s + 24·47-s + 21·49-s − 12·51-s − 12·57-s + 6·59-s + 14·61-s − 30·63-s + 18·67-s − 12·69-s + 6·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 2.26·7-s + 5/3·9-s − 0.603·11-s + 1.10·13-s + 1.45·17-s + 1.37·19-s + 2.61·21-s + 1.25·23-s − 4/5·25-s − 1.92·27-s + 1.11·29-s + 0.696·33-s + 0.328·37-s − 1.28·39-s + 0.937·41-s − 0.914·43-s + 3.50·47-s + 3·49-s − 1.68·51-s − 1.58·57-s + 0.781·59-s + 1.79·61-s − 3.77·63-s + 2.19·67-s − 1.44·69-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.299402127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.299402127\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 T - T^{2} - 2 T^{3} + 4 T^{4} - 2 p T^{5} - p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 + 6 T + 15 T^{2} + 6 p T^{3} + 20 p T^{4} + 6 p^{2} T^{5} + 15 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 2 T - T^{2} - 34 T^{3} - 140 T^{4} - 34 p T^{5} - p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 6 T + T^{2} - 6 T^{3} + 324 T^{4} - 6 p T^{5} + p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 6 T + 7 T^{2} + 54 T^{3} - 204 T^{4} + 54 p T^{5} + 7 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6 T - T^{2} + 54 T^{3} + 12 T^{4} + 54 p T^{5} - p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 6 T + T^{2} + 138 T^{3} - 660 T^{4} + 138 p T^{5} + p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 2 T + T^{2} + 142 T^{3} - 1508 T^{4} + 142 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 6 T - 47 T^{2} - 6 T^{3} + 3732 T^{4} - 6 p T^{5} - 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 6 T - 9 T^{2} - 246 T^{3} - 1372 T^{4} - 246 p T^{5} - 9 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6 T - 73 T^{2} + 54 T^{3} + 6276 T^{4} + 54 p T^{5} - 73 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 18 T + 127 T^{2} - 1134 T^{3} + 12612 T^{4} - 1134 p T^{5} + 127 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 6 T - 97 T^{2} + 54 T^{3} + 10092 T^{4} + 54 p T^{5} - 97 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 89 | $D_4\times C_2$ | \( 1 - 18 T + 97 T^{2} - 882 T^{3} + 14772 T^{4} - 882 p T^{5} + 97 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 9 T - 16 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.42367889985727221066954763027, −7.06463475797698566488590736097, −6.77347482949091301193423200549, −6.69773559035657252024482080805, −6.44605905808942323504532348242, −5.97729111066116453338391042122, −5.88759816220109391128941307254, −5.86287536364522807191495214106, −5.71351726054065319705523389686, −5.09881409939928174720054268498, −5.03117479839358553622332199153, −5.02576607554580027059582804928, −4.48367652267624510098683715905, −3.99108539997565107867849714579, −3.98873509949693845456780606314, −3.54334959693829717513962914932, −3.45918456745962179279567063873, −3.36195251581474341094222528916, −2.74608692525034609031094034858, −2.46564452129749095240690330058, −2.35876940303121592843404105727, −1.66137058791137089865050745018, −1.03454823071485268761463618883, −0.76067898754223866459061005006, −0.70595703498957099556870557272,
0.70595703498957099556870557272, 0.76067898754223866459061005006, 1.03454823071485268761463618883, 1.66137058791137089865050745018, 2.35876940303121592843404105727, 2.46564452129749095240690330058, 2.74608692525034609031094034858, 3.36195251581474341094222528916, 3.45918456745962179279567063873, 3.54334959693829717513962914932, 3.98873509949693845456780606314, 3.99108539997565107867849714579, 4.48367652267624510098683715905, 5.02576607554580027059582804928, 5.03117479839358553622332199153, 5.09881409939928174720054268498, 5.71351726054065319705523389686, 5.86287536364522807191495214106, 5.88759816220109391128941307254, 5.97729111066116453338391042122, 6.44605905808942323504532348242, 6.69773559035657252024482080805, 6.77347482949091301193423200549, 7.06463475797698566488590736097, 7.42367889985727221066954763027
