| L(s) = 1 | + 2·3-s + 4·5-s − 5·7-s + 9-s − 5·11-s + 13-s + 8·15-s + 4·17-s + 2·19-s − 10·21-s − 5·23-s + 10·25-s − 2·27-s − 5·29-s − 10·33-s − 20·35-s − 5·37-s + 2·39-s + 4·41-s + 4·43-s + 4·45-s + 18·47-s + 16·49-s + 8·51-s − 12·53-s − 20·55-s + 4·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s − 1.88·7-s + 1/3·9-s − 1.50·11-s + 0.277·13-s + 2.06·15-s + 0.970·17-s + 0.458·19-s − 2.18·21-s − 1.04·23-s + 2·25-s − 0.384·27-s − 0.928·29-s − 1.74·33-s − 3.38·35-s − 0.821·37-s + 0.320·39-s + 0.624·41-s + 0.609·43-s + 0.596·45-s + 2.62·47-s + 16/7·49-s + 1.12·51-s − 1.64·53-s − 2.69·55-s + 0.529·57-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
| \(L(1)\) |
\(\approx\) |
\(3.251267525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.251267525\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 5 T + 9 T^{2} + 10 T^{3} + 32 T^{4} + 10 p T^{5} + 9 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 5 T + T^{2} + 10 T^{3} + 180 T^{4} + 10 p T^{5} + p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 2 T - 18 T^{2} + 32 T^{3} + 47 T^{4} + 32 p T^{5} - 18 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 5 T - p T^{2} + 10 T^{3} + 1200 T^{4} + 10 p T^{5} - p^{3} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 5 T - T^{2} - 160 T^{3} - 890 T^{4} - 160 p T^{5} - p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 45 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 5 T - 51 T^{2} + 10 T^{3} + 3482 T^{4} + 10 p T^{5} - 51 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 4 T - 57 T^{2} + 52 T^{3} + 2792 T^{4} + 52 p T^{5} - 57 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 9 T + 110 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 5 T - 61 T^{2} - 160 T^{3} + 2440 T^{4} - 160 p T^{5} - 61 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 9 T - 23 T^{2} - 162 T^{3} + 2154 T^{4} - 162 p T^{5} - 23 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 9 T - 69 T^{2} - 144 T^{3} + 12584 T^{4} - 144 p T^{5} - 69 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 13 T + 184 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 18 T + 171 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 21 T + 141 T^{2} + 2226 T^{3} + 37814 T^{4} + 2226 p T^{5} + 141 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.90240083063285654779952894749, −6.36392088940078670100240173317, −6.23645939373602552606492170878, −5.90790909540550629794289029610, −5.89867677806275475236165706360, −5.76532717713122590832993036398, −5.55003029427577527971373316668, −5.40477755729558784142267734614, −4.95717949712662572665783131287, −4.88442959799443453249240801697, −4.40106488290895596183398120865, −4.32267322237057787874704424572, −3.81505388008590870843403028899, −3.75144431731700586124037469755, −3.42369503226892033697298010121, −3.19451522709642097795306839542, −2.97974188064362928323230407685, −2.74548988234364025493413856994, −2.56146943700815230721800551845, −2.23494200636166887441195859057, −2.07610397370672989004899809647, −1.67983438294814070596070400073, −1.33440431811206201798467641212, −0.821826898942895902373244117234, −0.31675666032518715080683484954,
0.31675666032518715080683484954, 0.821826898942895902373244117234, 1.33440431811206201798467641212, 1.67983438294814070596070400073, 2.07610397370672989004899809647, 2.23494200636166887441195859057, 2.56146943700815230721800551845, 2.74548988234364025493413856994, 2.97974188064362928323230407685, 3.19451522709642097795306839542, 3.42369503226892033697298010121, 3.75144431731700586124037469755, 3.81505388008590870843403028899, 4.32267322237057787874704424572, 4.40106488290895596183398120865, 4.88442959799443453249240801697, 4.95717949712662572665783131287, 5.40477755729558784142267734614, 5.55003029427577527971373316668, 5.76532717713122590832993036398, 5.89867677806275475236165706360, 5.90790909540550629794289029610, 6.23645939373602552606492170878, 6.36392088940078670100240173317, 6.90240083063285654779952894749
