| L(s) = 1 | + 4·2-s + 2·3-s + 8·4-s + 8·6-s + 8·8-s − 9-s − 10·11-s + 16·12-s + 8·13-s − 4·16-s − 6·17-s − 4·18-s + 10·19-s − 40·22-s + 12·23-s + 16·24-s − 9·25-s + 32·26-s − 2·27-s − 8·29-s − 4·31-s − 32·32-s − 20·33-s − 24·34-s − 8·36-s − 8·37-s + 40·38-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 1.15·3-s + 4·4-s + 3.26·6-s + 2.82·8-s − 1/3·9-s − 3.01·11-s + 4.61·12-s + 2.21·13-s − 16-s − 1.45·17-s − 0.942·18-s + 2.29·19-s − 8.52·22-s + 2.50·23-s + 3.26·24-s − 9/5·25-s + 6.27·26-s − 0.384·27-s − 1.48·29-s − 0.718·31-s − 5.65·32-s − 3.48·33-s − 4.11·34-s − 4/3·36-s − 1.31·37-s + 6.48·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.819250844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.819250844\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 T + 5 T^{2} - 10 T^{3} + 16 T^{4} - 10 p T^{5} + 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 + 10 T + 41 T^{2} + 82 T^{3} + 136 T^{4} + 82 p T^{5} + 41 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 120 T^{3} + 446 T^{4} - 120 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 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 | $D_4\times C_2$ | \( 1 - 10 T + 41 T^{2} - 66 T^{3} - 40 T^{4} - 66 p T^{5} + 41 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 105 T^{2} - 684 T^{3} + 3824 T^{4} - 684 p T^{5} + 105 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 248 T^{3} + 1918 T^{4} + 248 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 4 T - 47 T^{2} + 4 T^{3} + 2512 T^{4} + 4 p T^{5} - 47 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 12 T + 107 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \) |
| 41 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 3494 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 444 T^{3} + 2702 T^{4} - 444 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 12 T + 17 T^{2} + 396 T^{3} + 7152 T^{4} + 396 p T^{5} + 17 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 20 T + 101 T^{2} + 1072 T^{3} - 16076 T^{4} + 1072 p T^{5} + 101 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 26 T + 365 T^{2} + 3506 T^{3} + 28624 T^{4} + 3506 p T^{5} + 365 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 4 T + 53 T^{2} - 624 T^{3} + 1892 T^{4} - 624 p T^{5} + 53 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 6 T + 45 T^{2} - 738 T^{3} - 4576 T^{4} - 738 p T^{5} + 45 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 228 T^{2} + 22310 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 18 T + 277 T^{2} - 3042 T^{3} + 31116 T^{4} - 3042 p T^{5} + 277 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 16 T + 109 T^{2} + 176 T^{3} - 4856 T^{4} + 176 p T^{5} + 109 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 20 T + 200 T^{2} - 2180 T^{3} + 23086 T^{4} - 2180 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 9 T + 116 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−10.43898169690887556508257203536, −9.481796011209870622456345320148, −9.443874778242363644810683293079, −9.265692886026722219942833852813, −8.972245469777042854274095188035, −8.497196582956406258542520261480, −8.402782309814765828765097737070, −8.015743297465138467801449120467, −7.61626870263990658289046432823, −7.29849330920029166029439475402, −7.19444575880655847641669455476, −6.53939928113158581734795868057, −6.36897127787173522287939823319, −5.79298795266145422142016823081, −5.56632499461731680468894996119, −5.41386314944077732733944861136, −5.21586234007550947967474283453, −4.70514491732352094662851900158, −4.45794465664610703107600371842, −3.76213617381219925814576176267, −3.42177136837605976762229564734, −3.15932286533652217046788705721, −3.12933252734082075382580597383, −2.32140870906120728004430199755, −2.15882881959856427094114440845,
2.15882881959856427094114440845, 2.32140870906120728004430199755, 3.12933252734082075382580597383, 3.15932286533652217046788705721, 3.42177136837605976762229564734, 3.76213617381219925814576176267, 4.45794465664610703107600371842, 4.70514491732352094662851900158, 5.21586234007550947967474283453, 5.41386314944077732733944861136, 5.56632499461731680468894996119, 5.79298795266145422142016823081, 6.36897127787173522287939823319, 6.53939928113158581734795868057, 7.19444575880655847641669455476, 7.29849330920029166029439475402, 7.61626870263990658289046432823, 8.015743297465138467801449120467, 8.402782309814765828765097737070, 8.497196582956406258542520261480, 8.972245469777042854274095188035, 9.265692886026722219942833852813, 9.443874778242363644810683293079, 9.481796011209870622456345320148, 10.43898169690887556508257203536
