| L(s) = 1 | − 2·3-s + 4-s + 9-s − 2·12-s + 6·17-s + 12·19-s + 8·23-s − 4·25-s + 2·27-s − 16·29-s + 36-s − 24·37-s − 24·41-s + 2·43-s + 49-s − 12·51-s − 40·53-s − 24·57-s + 6·59-s − 64-s + 18·67-s + 6·68-s − 16·69-s + 24·71-s + 8·75-s + 12·76-s − 24·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 1/3·9-s − 0.577·12-s + 1.45·17-s + 2.75·19-s + 1.66·23-s − 4/5·25-s + 0.384·27-s − 2.97·29-s + 1/6·36-s − 3.94·37-s − 3.74·41-s + 0.304·43-s + 1/7·49-s − 1.68·51-s − 5.49·53-s − 3.17·57-s + 0.781·59-s − 1/8·64-s + 2.19·67-s + 0.727·68-s − 1.92·69-s + 2.84·71-s + 0.923·75-s + 1.37·76-s − 2.70·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7215346244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7215346244\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 6 T^{2} - 85 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 6 T + 5 T^{2} + 18 T^{3} + 60 T^{4} + 18 p T^{5} + 5 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 12 T + 82 T^{2} - 408 T^{3} + 1707 T^{4} - 408 p T^{5} + 82 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 8 T + 5 T^{2} - 104 T^{3} + 1480 T^{4} - 104 p T^{5} + 5 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 8 T + 35 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 82 T^{2} + 3171 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 24 T + 310 T^{2} + 2832 T^{3} + 19659 T^{4} + 2832 p T^{5} + 310 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 12 T + 89 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 2 T - 71 T^{2} + 22 T^{3} + 3604 T^{4} + 22 p T^{5} - 71 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 8006 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 20 T + 203 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6 T + 69 T^{2} - 342 T^{3} + 476 T^{4} - 342 p T^{5} + 69 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^3$ | \( 1 - 95 T^{2} + 5304 T^{4} - 95 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 18 T + 265 T^{2} - 2826 T^{3} + 27396 T^{4} - 2826 p T^{5} + 265 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 24 T + 357 T^{2} - 3960 T^{3} + 35816 T^{4} - 3960 p T^{5} + 357 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 14310 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 163 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 6 T + 93 T^{2} + 486 T^{3} - 292 T^{4} + 486 p T^{5} + 93 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 36 T + 730 T^{2} - 10728 T^{3} + 121299 T^{4} - 10728 p T^{5} + 730 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.69142391746753347386998246056, −7.56608541158320025189730485218, −7.09334399883042678238062887757, −6.93224731100799593607005033012, −6.91557202736186785793798584034, −6.89939411990615936289009071744, −6.29043475376490839500308371656, −5.95253303556012599295410803858, −5.89574126155553846202212122519, −5.34705187045503085558831304047, −5.33227071454830488358601285270, −5.26073625750644198636961068661, −4.95476999836256925769902203774, −4.88049162606403145839739213054, −4.31654837945409076956688125335, −3.61383730384312203796128503229, −3.58100304210979593434574221023, −3.40070455090560381716613569849, −3.14827916740683880830491384810, −3.04019460697396748757509041738, −2.07883979412458789837696819307, −1.90383509974229487257351589042, −1.45735749955846818859094979571, −1.26226376673888705563999383917, −0.29548496073519443075110509941,
0.29548496073519443075110509941, 1.26226376673888705563999383917, 1.45735749955846818859094979571, 1.90383509974229487257351589042, 2.07883979412458789837696819307, 3.04019460697396748757509041738, 3.14827916740683880830491384810, 3.40070455090560381716613569849, 3.58100304210979593434574221023, 3.61383730384312203796128503229, 4.31654837945409076956688125335, 4.88049162606403145839739213054, 4.95476999836256925769902203774, 5.26073625750644198636961068661, 5.33227071454830488358601285270, 5.34705187045503085558831304047, 5.89574126155553846202212122519, 5.95253303556012599295410803858, 6.29043475376490839500308371656, 6.89939411990615936289009071744, 6.91557202736186785793798584034, 6.93224731100799593607005033012, 7.09334399883042678238062887757, 7.56608541158320025189730485218, 7.69142391746753347386998246056
