| L(s) = 1 | − 6·5-s − 6·7-s + 18·11-s − 14·13-s + 8·19-s − 30·23-s + 3·25-s − 6·29-s + 74·31-s + 36·35-s − 120·37-s + 138·41-s + 10·43-s − 174·47-s + 11·49-s − 108·55-s + 18·59-s − 62·61-s + 84·65-s − 22·67-s + 40·73-s − 108·77-s − 86·79-s + 66·83-s + 84·91-s − 48·95-s + 242·97-s + ⋯ |
| L(s) = 1 | − 6/5·5-s − 6/7·7-s + 1.63·11-s − 1.07·13-s + 8/19·19-s − 1.30·23-s + 3/25·25-s − 0.206·29-s + 2.38·31-s + 1.02·35-s − 3.24·37-s + 3.36·41-s + 0.232·43-s − 3.70·47-s + 0.224·49-s − 1.96·55-s + 0.305·59-s − 1.01·61-s + 1.29·65-s − 0.328·67-s + 0.547·73-s − 1.40·77-s − 1.08·79-s + 0.795·83-s + 0.923·91-s − 0.505·95-s + 2.49·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7735498771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7735498771\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 6 T + 33 T^{2} + 126 T^{3} + 116 T^{4} + 126 p^{2} T^{5} + 33 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 6 T + 25 T^{2} - 522 T^{3} - 4044 T^{4} - 522 p^{2} T^{5} + 25 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 18 T + 249 T^{2} - 2538 T^{3} + 18308 T^{4} - 2538 p^{2} T^{5} + 249 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 14 T - 95 T^{2} - 658 T^{3} + 22996 T^{4} - 658 p^{2} T^{5} - 95 p^{4} T^{6} + 14 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 516 T^{2} + 135302 T^{4} - 516 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 4 T + 18 p T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 30 T + 1401 T^{2} + 33030 T^{3} + 1091060 T^{4} + 33030 p^{2} T^{5} + 1401 p^{4} T^{6} + 30 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 6 T + 1409 T^{2} + 8382 T^{3} + 1254420 T^{4} + 8382 p^{2} T^{5} + 1409 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 74 T + 2281 T^{2} - 94202 T^{3} + 4022068 T^{4} - 94202 p^{2} T^{5} + 2281 p^{4} T^{6} - 74 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 60 T + 3254 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 69 T + 3268 T^{2} - 69 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 10 T - 2087 T^{2} + 15110 T^{3} + 1179268 T^{4} + 15110 p^{2} T^{5} - 2087 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 174 T + 16745 T^{2} + 1157622 T^{3} + 61675956 T^{4} + 1157622 p^{2} T^{5} + 16745 p^{4} T^{6} + 174 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 996 T^{2} - 9136858 T^{4} - 996 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 18 T + 6969 T^{2} - 123498 T^{3} + 35331908 T^{4} - 123498 p^{2} T^{5} + 6969 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 62 T - 4463 T^{2} + 53630 T^{3} + 40856884 T^{4} + 53630 p^{2} T^{5} - 4463 p^{4} T^{6} + 62 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 22 T + 985 T^{2} - 208538 T^{3} - 22072796 T^{4} - 208538 p^{2} T^{5} + 985 p^{4} T^{6} + 22 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 16452 T^{2} + 117605702 T^{4} - 16452 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 20 T + 7302 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 86 T - 2231 T^{2} - 245530 T^{3} + 7570612 T^{4} - 245530 p^{2} T^{5} - 2231 p^{4} T^{6} + 86 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 66 T + 9321 T^{2} - 519354 T^{3} + 24465668 T^{4} - 519354 p^{2} T^{5} + 9321 p^{4} T^{6} - 66 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 25924 T^{2} + 285535302 T^{4} - 25924 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 242 T + 25489 T^{2} - 3450194 T^{3} + 454397668 T^{4} - 3450194 p^{2} T^{5} + 25489 p^{4} T^{6} - 242 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−8.588548281022193822472016331604, −8.473779131179955191028532182639, −8.361124854234024615915743835274, −7.79486429105064905291758731884, −7.73506291897392706085812816964, −7.57596764183362166120128695492, −7.02207010351586479754816338957, −6.90655938291977709870789604798, −6.68589314274298376830796860914, −6.25264775233533371345073832526, −6.03562605672951683663904691059, −6.01602264883943329438804714068, −5.25799654441817375418930516350, −5.00995198927872568159379746450, −4.82646789318593325748221981739, −4.19252989300505559542051177619, −4.03746716771174982255033387541, −4.02655104471445201777606211562, −3.26921175411692204301333951809, −3.15919867959449680180291541181, −2.85595286146428653839038556505, −2.05002562006856253027040108194, −1.77643973216313871964612600047, −1.00169079006518252323017645352, −0.29061276427115326601380175746,
0.29061276427115326601380175746, 1.00169079006518252323017645352, 1.77643973216313871964612600047, 2.05002562006856253027040108194, 2.85595286146428653839038556505, 3.15919867959449680180291541181, 3.26921175411692204301333951809, 4.02655104471445201777606211562, 4.03746716771174982255033387541, 4.19252989300505559542051177619, 4.82646789318593325748221981739, 5.00995198927872568159379746450, 5.25799654441817375418930516350, 6.01602264883943329438804714068, 6.03562605672951683663904691059, 6.25264775233533371345073832526, 6.68589314274298376830796860914, 6.90655938291977709870789604798, 7.02207010351586479754816338957, 7.57596764183362166120128695492, 7.73506291897392706085812816964, 7.79486429105064905291758731884, 8.361124854234024615915743835274, 8.473779131179955191028532182639, 8.588548281022193822472016331604
