| L(s) = 1 | − 2·3-s + 2·5-s + 5·9-s + 2·11-s + 4·13-s − 4·15-s − 2·17-s − 4·19-s − 4·23-s + 25-s − 10·27-s + 4·29-s − 12·31-s − 4·33-s − 4·37-s − 8·39-s + 24·41-s − 24·43-s + 10·45-s − 18·47-s + 4·51-s − 16·53-s + 4·55-s + 8·57-s − 12·59-s + 8·61-s + 8·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 5/3·9-s + 0.603·11-s + 1.10·13-s − 1.03·15-s − 0.485·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s − 1.92·27-s + 0.742·29-s − 2.15·31-s − 0.696·33-s − 0.657·37-s − 1.28·39-s + 3.74·41-s − 3.65·43-s + 1.49·45-s − 2.62·47-s + 0.560·51-s − 2.19·53-s + 0.539·55-s + 1.05·57-s − 1.56·59-s + 1.02·61-s + 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.006300856196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.006300856196\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 T - T^{2} - 2 T^{3} + 4 T^{4} - 2 p T^{5} - p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 - 2 T + 25 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 2 T - 29 T^{2} - 2 T^{3} + 732 T^{4} - 2 p T^{5} - 29 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 4 T - 16 T^{2} - 56 T^{3} + 127 T^{4} - 56 p T^{5} - 16 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 31 | $D_4\times C_2$ | \( 1 + 12 T + 64 T^{2} + 216 T^{3} + 975 T^{4} + 216 p T^{5} + 64 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T + 36 T^{2} - 376 T^{3} - 1561 T^{4} - 376 p T^{5} + 36 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 12 T + 116 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 18 T + 151 T^{2} + 1422 T^{3} + 12492 T^{4} + 1422 p T^{5} + 151 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 16 T + 88 T^{2} + 992 T^{3} + 11847 T^{4} + 992 p T^{5} + 88 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 8 T^{2} + 216 T^{3} + 6519 T^{4} + 216 p T^{5} + 8 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 8 T - 66 T^{2} - 64 T^{3} + 9275 T^{4} - 64 p T^{5} - 66 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^3$ | \( 1 - 132 T^{2} + 12935 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 16 T + 54 T^{2} - 896 T^{3} + 16787 T^{4} - 896 p T^{5} + 54 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 18 T + 117 T^{2} - 882 T^{3} + 11012 T^{4} - 882 p T^{5} + 117 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 8 T - 98 T^{2} - 128 T^{3} + 12627 T^{4} - 128 p T^{5} - 98 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 14 T + 193 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.44497068591366004737415805289, −6.29398014168821624788690407880, −6.19079110709963426876900338950, −6.08545876644797208834402016467, −5.64763767809327147424584274986, −5.57010975786192793271425688755, −5.24285683449288367217018594545, −4.96086590701294201346008071561, −4.90643311536990496119454909788, −4.74472263817793747892862102571, −4.35771679794329488623745201294, −4.13456675316856156512176725016, −4.08485258834485946496757842160, −3.55972982862249453686778384503, −3.36813422973163646227428615269, −3.29811471289852055506978200965, −3.26983867463519943360798286105, −2.32843151174142200171354169124, −2.13119956313895907924894190081, −2.03795226794678547387537672081, −1.95598400487570619758314104402, −1.36494022191740820405549841537, −1.21247644087052288830120455479, −0.855584788905528014382787503215, −0.01514302682623437574153683483,
0.01514302682623437574153683483, 0.855584788905528014382787503215, 1.21247644087052288830120455479, 1.36494022191740820405549841537, 1.95598400487570619758314104402, 2.03795226794678547387537672081, 2.13119956313895907924894190081, 2.32843151174142200171354169124, 3.26983867463519943360798286105, 3.29811471289852055506978200965, 3.36813422973163646227428615269, 3.55972982862249453686778384503, 4.08485258834485946496757842160, 4.13456675316856156512176725016, 4.35771679794329488623745201294, 4.74472263817793747892862102571, 4.90643311536990496119454909788, 4.96086590701294201346008071561, 5.24285683449288367217018594545, 5.57010975786192793271425688755, 5.64763767809327147424584274986, 6.08545876644797208834402016467, 6.19079110709963426876900338950, 6.29398014168821624788690407880, 6.44497068591366004737415805289
