| L(s) = 1 | + 2·2-s + 6·3-s + 4-s − 3·5-s + 12·6-s + 3·7-s − 2·8-s + 21·9-s − 6·10-s − 10·11-s + 6·12-s + 6·14-s − 18·15-s − 4·16-s + 17-s + 42·18-s + 15·19-s − 3·20-s + 18·21-s − 20·22-s − 3·23-s − 12·24-s + 54·27-s + 3·28-s + 4·29-s − 36·30-s + 13·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3.46·3-s + 1/2·4-s − 1.34·5-s + 4.89·6-s + 1.13·7-s − 0.707·8-s + 7·9-s − 1.89·10-s − 3.01·11-s + 1.73·12-s + 1.60·14-s − 4.64·15-s − 16-s + 0.242·17-s + 9.89·18-s + 3.44·19-s − 0.670·20-s + 3.92·21-s − 4.26·22-s − 0.625·23-s − 2.44·24-s + 10.3·27-s + 0.566·28-s + 0.742·29-s − 6.57·30-s + 2.33·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(15.81879832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.81879832\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + p T^{2} )^{2}( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} ) \) |
| 13 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 126 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - T - 19 T^{2} + 14 T^{3} + 94 T^{4} + 14 p T^{5} - 19 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 15 T + 127 T^{2} - 780 T^{3} + 3768 T^{4} - 780 p T^{5} + 127 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 3 T + 7 T^{2} + 12 T^{3} - 444 T^{4} + 12 p T^{5} + 7 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 31 | $D_4\times C_2$ | \( 1 - 13 T + 79 T^{2} - 364 T^{3} + 1900 T^{4} - 364 p T^{5} + 79 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 7 T - 23 T^{2} + 14 T^{3} + 2002 T^{4} + 14 p T^{5} - 23 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 2 T + 26 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 41 T^{2} + 2964 T^{4} - 41 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 15 T + 145 T^{2} - 1050 T^{3} + 6216 T^{4} - 1050 p T^{5} + 145 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 9 T + 21 T^{2} - 54 T^{3} - 1342 T^{4} - 54 p T^{5} + 21 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 3 T + 62 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2 T - 74 T^{2} + 112 T^{3} + 1327 T^{4} + 112 p T^{5} - 74 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 153 T^{2} + 14780 T^{4} - 153 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 12 T + 130 T^{2} + 984 T^{3} + 4899 T^{4} + 984 p T^{5} + 130 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 15 T + 247 T^{2} - 2580 T^{3} + 29268 T^{4} - 2580 p T^{5} + 247 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 13 T + 194 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 6 T + 174 T^{2} - 972 T^{3} + 19391 T^{4} - 972 p T^{5} + 174 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 12 T + 173 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.942126861515934807031836336918, −7.81942242233366343272188811582, −7.63027171604852118117535807468, −7.62238238428164701709358103708, −7.21914506834317024620031830564, −7.09340396183370990095555520590, −6.65874049785282579392640983858, −6.22463520039020577405745043501, −5.77180353286523159328634888180, −5.56511363523310382330133237298, −5.40777516291165696685786348317, −4.83136391387339151168798874534, −4.81975741474389743653839110795, −4.60663407343072318202210633098, −4.41799023783849148920457403516, −3.76848694640105692047849025268, −3.73658488115899998778219377053, −3.47810205278236330898581687146, −3.30920562974616099709898504477, −2.75323262238177310536959935458, −2.50248424806857233177714434695, −2.47021354102504723268699959946, −2.35357867605031612699881086198, −1.23777726211186728535550984919, −0.998232693544978054826705476656,
0.998232693544978054826705476656, 1.23777726211186728535550984919, 2.35357867605031612699881086198, 2.47021354102504723268699959946, 2.50248424806857233177714434695, 2.75323262238177310536959935458, 3.30920562974616099709898504477, 3.47810205278236330898581687146, 3.73658488115899998778219377053, 3.76848694640105692047849025268, 4.41799023783849148920457403516, 4.60663407343072318202210633098, 4.81975741474389743653839110795, 4.83136391387339151168798874534, 5.40777516291165696685786348317, 5.56511363523310382330133237298, 5.77180353286523159328634888180, 6.22463520039020577405745043501, 6.65874049785282579392640983858, 7.09340396183370990095555520590, 7.21914506834317024620031830564, 7.62238238428164701709358103708, 7.63027171604852118117535807468, 7.81942242233366343272188811582, 7.942126861515934807031836336918
