| L(s) = 1 | + 2·2-s + 4-s + 4·5-s − 2·8-s + 8·10-s − 8·11-s − 4·16-s − 4·17-s + 10·19-s + 4·20-s − 16·22-s + 4·23-s + 2·25-s − 4·29-s + 12·31-s − 2·32-s − 8·34-s − 4·37-s + 20·38-s − 8·40-s − 4·43-s − 8·44-s + 8·46-s + 4·50-s − 12·53-s − 32·55-s − 8·58-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s + 1.78·5-s − 0.707·8-s + 2.52·10-s − 2.41·11-s − 16-s − 0.970·17-s + 2.29·19-s + 0.894·20-s − 3.41·22-s + 0.834·23-s + 2/5·25-s − 0.742·29-s + 2.15·31-s − 0.353·32-s − 1.37·34-s − 0.657·37-s + 3.24·38-s − 1.26·40-s − 0.609·43-s − 1.20·44-s + 1.17·46-s + 0.565·50-s − 1.64·53-s − 4.31·55-s − 1.05·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06372122555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06372122555\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( ( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 - 2 T^{2} - 165 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 10 T + 43 T^{2} - 10 p T^{3} + 52 p T^{4} - 10 p^{2} T^{5} + 43 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 29 | $D_4\times C_2$ | \( 1 + 4 T - 22 T^{2} - 80 T^{3} + 139 T^{4} - 80 p T^{5} - 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 6 T + 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T + 34 T^{2} - 368 T^{3} - 1637 T^{4} - 368 p T^{5} + 34 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 14 T^{2} - 1485 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T - 50 T^{2} - 80 T^{3} + 1819 T^{4} - 80 p T^{5} - 50 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 2 T^{2} - 2205 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 12 T + 26 T^{2} + 144 T^{3} + 3483 T^{4} + 144 p T^{5} + 26 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 2 T - 55 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 18 T + 127 T^{2} - 1350 T^{3} + 15324 T^{4} - 1350 p T^{5} + 127 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 16 T + 82 T^{2} - 640 T^{3} + 8635 T^{4} - 640 p T^{5} + 82 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 10 T + 143 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 4 T - 110 T^{2} - 80 T^{3} + 9379 T^{4} - 80 p T^{5} - 110 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 6 T - 107 T^{2} - 90 T^{3} + 11364 T^{4} - 90 p T^{5} - 107 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 2 T - 79 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 24 T + 278 T^{2} - 2880 T^{3} + 29619 T^{4} - 2880 p T^{5} + 278 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 4 T - 158 T^{2} - 80 T^{3} + 19315 T^{4} - 80 p T^{5} - 158 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.39085839089398548989726409934, −5.77764622254140858511673870107, −5.69747995975721397881395821644, −5.60797095831921694116980187079, −5.38271771421228505954457631872, −5.20305565190789014759817952038, −5.18651390174391905444547937727, −5.05321508738326372343903427207, −4.79369042744480428501256279366, −4.41186245535013464796791299862, −4.11013448468002165511807373505, −4.07716418972361414240284614163, −3.87134625458873143604900325432, −3.49274214956498256624178672023, −3.14844539188766627612044599532, −2.92862015896377063784862479057, −2.75324408627812732753980893456, −2.60633314597052510229601298366, −2.57793306275771064184363887089, −2.10842928785853012845684540391, −1.80945332153544940004552005197, −1.35394396874967684758053898140, −1.34359875876115205011525392888, −0.78372313184061596955357122483, −0.02795205170989524250949000002,
0.02795205170989524250949000002, 0.78372313184061596955357122483, 1.34359875876115205011525392888, 1.35394396874967684758053898140, 1.80945332153544940004552005197, 2.10842928785853012845684540391, 2.57793306275771064184363887089, 2.60633314597052510229601298366, 2.75324408627812732753980893456, 2.92862015896377063784862479057, 3.14844539188766627612044599532, 3.49274214956498256624178672023, 3.87134625458873143604900325432, 4.07716418972361414240284614163, 4.11013448468002165511807373505, 4.41186245535013464796791299862, 4.79369042744480428501256279366, 5.05321508738326372343903427207, 5.18651390174391905444547937727, 5.20305565190789014759817952038, 5.38271771421228505954457631872, 5.60797095831921694116980187079, 5.69747995975721397881395821644, 5.77764622254140858511673870107, 6.39085839089398548989726409934
