| L(s) = 1 | + 2·2-s + 2·3-s + 4-s + 4·6-s + 3·7-s − 2·8-s + 9-s + 2·12-s + 13-s + 6·14-s − 4·16-s + 2·17-s + 2·18-s + 3·19-s + 6·21-s − 3·23-s − 4·24-s + 2·26-s − 2·27-s + 3·28-s − 9·29-s − 2·31-s − 2·32-s + 4·34-s + 36-s + 6·38-s + 2·39-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 1/2·4-s + 1.63·6-s + 1.13·7-s − 0.707·8-s + 1/3·9-s + 0.577·12-s + 0.277·13-s + 1.60·14-s − 16-s + 0.485·17-s + 0.471·18-s + 0.688·19-s + 1.30·21-s − 0.625·23-s − 0.816·24-s + 0.392·26-s − 0.384·27-s + 0.566·28-s − 1.67·29-s − 0.359·31-s − 0.353·32-s + 0.685·34-s + 1/6·36-s + 0.973·38-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.694083217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.694083217\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 3 T - 3 T^{2} + 6 T^{3} + 32 T^{4} + 6 p T^{5} - 3 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} )( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $D_4\times C_2$ | \( 1 - 2 T - 14 T^{2} + 32 T^{3} - 33 T^{4} + 32 p T^{5} - 14 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 3 T - 27 T^{2} + 6 T^{3} + 764 T^{4} + 6 p T^{5} - 27 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 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 | $D_4\times C_2$ | \( 1 + 9 T + 7 T^{2} + 144 T^{3} + 2286 T^{4} + 144 p T^{5} + 7 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + T + 24 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 - 57 T^{2} + 1880 T^{4} - 57 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 8 T + 34 T^{2} - 416 T^{3} - 3405 T^{4} - 416 p T^{5} + 34 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 5 T - 63 T^{2} - 10 T^{3} + 4820 T^{4} - 10 p T^{5} - 63 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 53 | $D_{4}$ | \( ( 1 + 13 T + 144 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 17 T + 103 T^{2} + 1156 T^{3} + 14064 T^{4} + 1156 p T^{5} + 103 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12 T + 42 T^{2} + 384 T^{3} - 3733 T^{4} + 384 p T^{5} + 42 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 18 T + 118 T^{2} - 1152 T^{3} + 14391 T^{4} - 1152 p T^{5} + 118 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 6 T + 2 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 19 T + 244 T^{2} - 19 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 6 T + 158 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 17 T + 145 T^{2} + 578 T^{3} - 12906 T^{4} + 578 p T^{5} + 145 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 6 T - 150 T^{2} - 48 T^{3} + 21695 T^{4} - 48 p T^{5} - 150 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.39815242730123495962426256673, −6.38084831624460273275557987701, −6.16811977613832819162866277202, −5.92005910430781055589221638894, −5.59506289509469387741373198455, −5.18842115578722503515741826327, −5.07665407213602374863475260071, −4.99096248255018624432697503198, −4.97401820693630656476477889974, −4.75433206836108902567565364406, −4.42834041583236543987221232228, −3.92698904711575687475767510934, −3.76915656329082789458868691119, −3.76474908758081342739552066432, −3.39058646748494261182910691169, −3.32331371544325126522387691653, −3.22044603385700993130486374991, −2.61496792528600889440224288682, −2.56809119830715925666651768577, −2.06246911768661774385854681975, −1.93544158199276962788318377983, −1.70171221590461482838926283432, −1.32785401268334758728197865330, −0.979687609296254591784940691191, −0.13747727961320575846342534922,
0.13747727961320575846342534922, 0.979687609296254591784940691191, 1.32785401268334758728197865330, 1.70171221590461482838926283432, 1.93544158199276962788318377983, 2.06246911768661774385854681975, 2.56809119830715925666651768577, 2.61496792528600889440224288682, 3.22044603385700993130486374991, 3.32331371544325126522387691653, 3.39058646748494261182910691169, 3.76474908758081342739552066432, 3.76915656329082789458868691119, 3.92698904711575687475767510934, 4.42834041583236543987221232228, 4.75433206836108902567565364406, 4.97401820693630656476477889974, 4.99096248255018624432697503198, 5.07665407213602374863475260071, 5.18842115578722503515741826327, 5.59506289509469387741373198455, 5.92005910430781055589221638894, 6.16811977613832819162866277202, 6.38084831624460273275557987701, 6.39815242730123495962426256673
