| L(s) = 1 | − 4·2-s + 10·4-s − 4·5-s + 4·7-s − 20·8-s + 16·10-s − 2·11-s − 3·13-s − 16·14-s + 35·16-s − 6·17-s + 3·19-s − 40·20-s + 8·22-s + 23-s + 10·25-s + 12·26-s + 40·28-s + 3·29-s + 3·31-s − 56·32-s + 24·34-s − 16·35-s − 4·37-s − 12·38-s + 80·40-s + 11·41-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 5·4-s − 1.78·5-s + 1.51·7-s − 7.07·8-s + 5.05·10-s − 0.603·11-s − 0.832·13-s − 4.27·14-s + 35/4·16-s − 1.45·17-s + 0.688·19-s − 8.94·20-s + 1.70·22-s + 0.208·23-s + 2·25-s + 2.35·26-s + 7.55·28-s + 0.557·29-s + 0.538·31-s − 9.89·32-s + 4.11·34-s − 2.70·35-s − 0.657·37-s − 1.94·38-s + 12.6·40-s + 1.71·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8027854582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8027854582\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| 37 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 7 | $C_2 \wr S_4$ | \( 1 - 4 T + 15 T^{2} - 44 T^{3} + 96 T^{4} - 44 p T^{5} + 15 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 2 T + 21 T^{2} - 2 T^{3} + 180 T^{4} - 2 p T^{5} + 21 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 3 T + 14 T^{2} + 73 T^{3} + 354 T^{4} + 73 p T^{5} + 14 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 6 T + 21 T^{2} - 106 T^{3} - 628 T^{4} - 106 p T^{5} + 21 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 3 T + 26 T^{2} + 37 T^{3} + 186 T^{4} + 37 p T^{5} + 26 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - T + 16 T^{2} - 5 T^{3} + 958 T^{4} - 5 p T^{5} + 16 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 3 T + 34 T^{2} - 57 T^{3} + 1498 T^{4} - 57 p T^{5} + 34 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 3 T + 86 T^{2} - 9 p T^{3} + 3442 T^{4} - 9 p^{2} T^{5} + 86 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 11 T + 168 T^{2} - 1269 T^{3} + 10334 T^{4} - 1269 p T^{5} + 168 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 5 T + 96 T^{2} + 517 T^{3} + 5838 T^{4} + 517 p T^{5} + 96 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 100 T^{2} + 96 T^{3} + 6262 T^{4} + 96 p T^{5} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 22 T + 333 T^{2} - 3254 T^{3} + 27156 T^{4} - 3254 p T^{5} + 333 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 8 T + 184 T^{2} - 1096 T^{3} + 14494 T^{4} - 1096 p T^{5} + 184 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 5 T + 114 T^{2} + 1111 T^{3} + 7562 T^{4} + 1111 p T^{5} + 114 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 6 T + 196 T^{2} - 1174 T^{3} + 17414 T^{4} - 1174 p T^{5} + 196 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 18 T + 228 T^{2} - 2298 T^{3} + 22806 T^{4} - 2298 p T^{5} + 228 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 5 T + 164 T^{2} - 37 T^{3} + 150 p T^{4} - 37 p T^{5} + 164 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - T + 98 T^{2} - 953 T^{3} + 7834 T^{4} - 953 p T^{5} + 98 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 5 T + 122 T^{2} + 621 T^{3} + 2666 T^{4} + 621 p T^{5} + 122 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 7 T + 152 T^{2} - 1697 T^{3} + 20014 T^{4} - 1697 p T^{5} + 152 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.29410643270562873963651650926, −6.10394814717993867156852841037, −5.71191289886424506655659181555, −5.45296833222026870746612625622, −5.41305324396109122843952938560, −5.07612099289584404579755987640, −4.88844365563967611561423287689, −4.75211608781613119658309285907, −4.50673064796484224749831469050, −4.24082500344435270043478823358, −4.02653694099066377421803355165, −3.72130373609701099854069605820, −3.63064946609788112375752761359, −3.15747580856256755274545797900, −2.97744751594233946518807911358, −2.90803013972299712642272519845, −2.43910836535835245514511280788, −2.22118524646101579447834432215, −2.11943939424732675601966374182, −1.76626570129276185306923217313, −1.70365065409675702201901427759, −0.882877305450239840656309263477, −0.833607448087418395372710935486, −0.69168407326170216395107205197, −0.34693216058570296742577216286,
0.34693216058570296742577216286, 0.69168407326170216395107205197, 0.833607448087418395372710935486, 0.882877305450239840656309263477, 1.70365065409675702201901427759, 1.76626570129276185306923217313, 2.11943939424732675601966374182, 2.22118524646101579447834432215, 2.43910836535835245514511280788, 2.90803013972299712642272519845, 2.97744751594233946518807911358, 3.15747580856256755274545797900, 3.63064946609788112375752761359, 3.72130373609701099854069605820, 4.02653694099066377421803355165, 4.24082500344435270043478823358, 4.50673064796484224749831469050, 4.75211608781613119658309285907, 4.88844365563967611561423287689, 5.07612099289584404579755987640, 5.41305324396109122843952938560, 5.45296833222026870746612625622, 5.71191289886424506655659181555, 6.10394814717993867156852841037, 6.29410643270562873963651650926
