| L(s) = 1 | − 2·2-s − 3-s + 4-s + 2·6-s − 10·7-s + 2·8-s + 2·9-s + 4·11-s − 12-s − 4·13-s + 20·14-s − 4·16-s + 2·17-s − 4·18-s − 10·19-s + 10·21-s − 8·22-s + 3·23-s − 2·24-s + 8·26-s + 27-s − 10·28-s − 4·29-s + 8·31-s + 2·32-s − 4·33-s − 4·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 1/2·4-s + 0.816·6-s − 3.77·7-s + 0.707·8-s + 2/3·9-s + 1.20·11-s − 0.288·12-s − 1.10·13-s + 5.34·14-s − 16-s + 0.485·17-s − 0.942·18-s − 2.29·19-s + 2.18·21-s − 1.70·22-s + 0.625·23-s − 0.408·24-s + 1.56·26-s + 0.192·27-s − 1.88·28-s − 0.742·29-s + 1.43·31-s + 0.353·32-s − 0.696·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1891439667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1891439667\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + T - T^{2} - 4 T^{3} - 8 T^{4} - 4 p T^{5} - p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 5 T + 16 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2 T - 14 T^{2} + 32 T^{3} - 33 T^{4} + 32 p T^{5} - 14 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 3 T - T^{2} + 108 T^{3} - 636 T^{4} + 108 p T^{5} - p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 3 T + 38 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 4 T - 53 T^{2} + 52 T^{3} + 2424 T^{4} + 52 p T^{5} - 53 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 2 T - 66 T^{2} - 32 T^{3} + 2879 T^{4} - 32 p T^{5} - 66 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 14 T + 70 T^{2} - 448 T^{3} + 4455 T^{4} - 448 p T^{5} + 70 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 11 T - 11 T^{2} + 286 T^{3} + 8202 T^{4} + 286 p T^{5} - 11 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 25 T + 355 T^{2} + 3800 T^{3} + 32544 T^{4} + 3800 p T^{5} + 355 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 2 T - 102 T^{2} - 32 T^{3} + 7271 T^{4} - 32 p T^{5} - 102 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 23 T + 267 T^{2} - 2944 T^{3} + 28712 T^{4} - 2944 p T^{5} + 267 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 16 T + 118 T^{2} - 64 T^{3} - 4173 T^{4} - 64 p T^{5} + 118 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 9 T - 47 T^{2} + 162 T^{3} + 5142 T^{4} + 162 p T^{5} - 47 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 14 T + 6 T^{2} - 448 T^{3} + 14375 T^{4} - 448 p T^{5} + 6 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - T + 60 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 7 T - 103 T^{2} - 182 T^{3} + 10822 T^{4} - 182 p T^{5} - 103 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 9 T - 95 T^{2} - 162 T^{3} + 13710 T^{4} - 162 p T^{5} - 95 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.03647425469056582704782230544, −6.87079552181877487720511514253, −6.77053026046541362276791794488, −6.50339896577056259295196510060, −6.40431852113881690202360225857, −6.30171102648632785069654370486, −6.09619932431761870005228223442, −5.73845687576166860524115295903, −5.48152781005895761180796559248, −5.09561205512416389199311598435, −4.82585311783983490856143275476, −4.67860278308229281341404030444, −4.22050970552036277765789174689, −4.01972636108861591190358693220, −3.96314801014182569624082848351, −3.47549000116529370054881429046, −3.32582065808810436017132429337, −2.92781513086333304502025845075, −2.79327854421939094898649892545, −2.39325407438640905907261968203, −2.03685003497668402038170409333, −1.54026414417845786939621825360, −1.16614741241938070408549416878, −0.49198785928873035586696560899, −0.29114173388607204644534243293,
0.29114173388607204644534243293, 0.49198785928873035586696560899, 1.16614741241938070408549416878, 1.54026414417845786939621825360, 2.03685003497668402038170409333, 2.39325407438640905907261968203, 2.79327854421939094898649892545, 2.92781513086333304502025845075, 3.32582065808810436017132429337, 3.47549000116529370054881429046, 3.96314801014182569624082848351, 4.01972636108861591190358693220, 4.22050970552036277765789174689, 4.67860278308229281341404030444, 4.82585311783983490856143275476, 5.09561205512416389199311598435, 5.48152781005895761180796559248, 5.73845687576166860524115295903, 6.09619932431761870005228223442, 6.30171102648632785069654370486, 6.40431852113881690202360225857, 6.50339896577056259295196510060, 6.77053026046541362276791794488, 6.87079552181877487720511514253, 7.03647425469056582704782230544
