| L(s) = 1 | + 2·3-s − 4·5-s − 2·7-s + 9-s + 4·11-s + 6·13-s − 8·15-s + 8·17-s − 4·19-s − 4·21-s + 4·23-s + 10·25-s − 2·27-s + 8·29-s − 12·31-s + 8·33-s + 8·35-s − 4·37-s + 12·39-s − 4·41-s − 14·43-s − 4·45-s + 8·47-s + 5·49-s + 16·51-s − 16·55-s − 8·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s − 0.755·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s − 2.06·15-s + 1.94·17-s − 0.917·19-s − 0.872·21-s + 0.834·23-s + 2·25-s − 0.384·27-s + 1.48·29-s − 2.15·31-s + 1.39·33-s + 1.35·35-s − 0.657·37-s + 1.92·39-s − 0.624·41-s − 2.13·43-s − 0.596·45-s + 1.16·47-s + 5/7·49-s + 2.24·51-s − 2.15·55-s − 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.999663534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.999663534\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 2 T - T^{2} - 18 T^{3} - 52 T^{4} - 18 p T^{5} - p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 8 T + 24 T^{2} - 48 T^{3} + 223 T^{4} - 48 p T^{5} + 24 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 8 T - 48 T^{3} + 1399 T^{4} - 48 p T^{5} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 6 T + p T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T - 22 T^{2} - 144 T^{3} - 517 T^{4} - 144 p T^{5} - 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 4 T - 60 T^{2} - 24 T^{3} + 3439 T^{4} - 24 p T^{5} - 60 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 14 T + 71 T^{2} + 546 T^{3} + 5348 T^{4} + 546 p T^{5} + 71 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 8 T - 60 T^{2} - 48 T^{3} + 8119 T^{4} - 48 p T^{5} - 60 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 14 T + 23 T^{2} + 546 T^{3} + 12308 T^{4} + 546 p T^{5} + 23 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 8 T - 4 T^{2} + 592 T^{3} - 4961 T^{4} + 592 p T^{5} - 4 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 6 T + 145 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12 T + 20 T^{2} + 648 T^{3} - 5361 T^{4} + 648 p T^{5} + 20 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 6 T + 83 T^{2} - 1446 T^{3} - 10692 T^{4} - 1446 p T^{5} + 83 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.68552624759031894594561701024, −6.49744251544911036933718549654, −6.41210480003247408956033606435, −6.16572373340526622370234517942, −5.99617148284501791290634440915, −5.66768955616960738957639243789, −5.42873300444480055101420537335, −5.12550873319890695795127753293, −4.89318094306222502417230133731, −4.69399344852792989210299597342, −4.44904808021920130465449129724, −4.14079687724193596619148296892, −3.73885815337088984485363399541, −3.71699076504449678176402542571, −3.61686718833651741303723888113, −3.23086969810478083994888703164, −3.21057665565191103013764202998, −3.12141546773569137141680512959, −2.58821254394243041194366465828, −2.21004397976254131533640041590, −1.81818417510931104273878542811, −1.58113463419475597125381873997, −1.15032272715632732947321871745, −0.860751371202008362204705669256, −0.35064554229325580389145724770,
0.35064554229325580389145724770, 0.860751371202008362204705669256, 1.15032272715632732947321871745, 1.58113463419475597125381873997, 1.81818417510931104273878542811, 2.21004397976254131533640041590, 2.58821254394243041194366465828, 3.12141546773569137141680512959, 3.21057665565191103013764202998, 3.23086969810478083994888703164, 3.61686718833651741303723888113, 3.71699076504449678176402542571, 3.73885815337088984485363399541, 4.14079687724193596619148296892, 4.44904808021920130465449129724, 4.69399344852792989210299597342, 4.89318094306222502417230133731, 5.12550873319890695795127753293, 5.42873300444480055101420537335, 5.66768955616960738957639243789, 5.99617148284501791290634440915, 6.16572373340526622370234517942, 6.41210480003247408956033606435, 6.49744251544911036933718549654, 6.68552624759031894594561701024
