| L(s) = 1 | + 4·2-s − 4·3-s + 8·4-s + 8·5-s − 16·6-s + 16·8-s + 8·9-s + 32·10-s − 32·12-s − 32·15-s + 36·16-s + 16·17-s + 32·18-s − 16·19-s + 64·20-s + 4·23-s − 64·24-s + 32·25-s − 12·27-s − 128·30-s − 8·31-s + 64·32-s + 64·34-s + 64·36-s − 20·37-s − 64·38-s + 128·40-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 2.30·3-s + 4·4-s + 3.57·5-s − 6.53·6-s + 5.65·8-s + 8/3·9-s + 10.1·10-s − 9.23·12-s − 8.26·15-s + 9·16-s + 3.88·17-s + 7.54·18-s − 3.67·19-s + 14.3·20-s + 0.834·23-s − 13.0·24-s + 32/5·25-s − 2.30·27-s − 23.3·30-s − 1.43·31-s + 11.3·32-s + 10.9·34-s + 32/3·36-s − 3.28·37-s − 10.3·38-s + 20.2·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.718898538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.718898538\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 96 T^{3} + 239 T^{4} - 96 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 82 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 19 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 672 T^{3} + 2999 T^{4} + 672 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 2 T + 39 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 50 T^{2} + 2179 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} - 80 T^{3} - 1481 T^{4} - 80 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 1660 T^{3} + 11662 T^{4} + 1660 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 592 T^{3} + 2722 T^{4} + 592 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 106 T^{2} + 6379 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 1120 T^{3} + 9271 T^{4} + 1120 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 46 T^{2} + 5635 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 + 6034 T^{4} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} - 200 T^{3} - 7214 T^{4} - 200 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 28 T + 392 T^{2} - 3724 T^{3} + 31534 T^{4} - 3724 p T^{5} + 392 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 576 T^{3} + 10367 T^{4} + 576 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 6 T + 159 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 13447 T^{4} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 32 T + 512 T^{2} - 6656 T^{3} + 72367 T^{4} - 6656 p T^{5} + 512 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 32 T + 512 T^{2} + 7168 T^{3} + 84223 T^{4} + 7168 p T^{5} + 512 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.710882544546766186518915557275, −8.359353342319508761321387458560, −7.955129482345064531248970639130, −7.64640171933969498915770182426, −7.53768224808193586969351667511, −6.80025269453675098597481583672, −6.71780752422484728308714635348, −6.69955002229246868456957204888, −6.21010915021482259424670481310, −6.11012138523813880005662364968, −5.90741961748669634049818250843, −5.35027148047421015330174439788, −5.33761085377088589800288187358, −5.26614995240576098841311492731, −5.23239027688109253442316178727, −4.77143753746381227798769931741, −4.60242689833863846093487179547, −3.99396257667187070314564170560, −3.63487589580332348181541940916, −3.34377727841273432338380767964, −3.13835238599765536614706366613, −2.06775714843674350674621483690, −1.93813770050050353877005677327, −1.61371997258903642255492442965, −1.40167872400662859160517586924,
1.40167872400662859160517586924, 1.61371997258903642255492442965, 1.93813770050050353877005677327, 2.06775714843674350674621483690, 3.13835238599765536614706366613, 3.34377727841273432338380767964, 3.63487589580332348181541940916, 3.99396257667187070314564170560, 4.60242689833863846093487179547, 4.77143753746381227798769931741, 5.23239027688109253442316178727, 5.26614995240576098841311492731, 5.33761085377088589800288187358, 5.35027148047421015330174439788, 5.90741961748669634049818250843, 6.11012138523813880005662364968, 6.21010915021482259424670481310, 6.69955002229246868456957204888, 6.71780752422484728308714635348, 6.80025269453675098597481583672, 7.53768224808193586969351667511, 7.64640171933969498915770182426, 7.955129482345064531248970639130, 8.359353342319508761321387458560, 8.710882544546766186518915557275
