| L(s) = 1 | − 3-s + 3·5-s + 2·7-s + 3·9-s − 4·13-s − 3·15-s + 6·17-s + 8·19-s − 2·21-s − 12·23-s + 5·25-s − 5·31-s + 6·35-s + 37-s + 4·39-s + 40·43-s + 9·45-s + 7·49-s − 6·51-s + 6·53-s − 8·57-s − 3·59-s − 4·61-s + 6·63-s − 12·65-s − 4·67-s + 12·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 0.755·7-s + 9-s − 1.10·13-s − 0.774·15-s + 1.45·17-s + 1.83·19-s − 0.436·21-s − 2.50·23-s + 25-s − 0.898·31-s + 1.01·35-s + 0.164·37-s + 0.640·39-s + 6.09·43-s + 1.34·45-s + 49-s − 0.840·51-s + 0.824·53-s − 1.05·57-s − 0.390·59-s − 0.512·61-s + 0.755·63-s − 1.48·65-s − 0.488·67-s + 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.595012652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.595012652\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 3 | $C_4\times C_2$ | \( 1 + T - 2 T^{2} - 5 T^{3} + T^{4} - 5 p T^{5} - 2 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_4\times C_2$ | \( 1 - 3 T + 4 T^{2} + 3 T^{3} - 29 T^{4} + 3 p T^{5} + 4 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 - 2 T - 3 T^{2} + 20 T^{3} - 19 T^{4} + 20 p T^{5} - 3 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_4\times C_2$ | \( 1 + 4 T + 3 T^{2} - 40 T^{3} - 199 T^{4} - 40 p T^{5} + 3 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 - 6 T + 19 T^{2} - 12 T^{3} - 251 T^{4} - 12 p T^{5} + 19 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 - 8 T + 45 T^{2} - 208 T^{3} + 809 T^{4} - 208 p T^{5} + 45 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 29 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 + 5 T - 6 T^{2} - 185 T^{3} - 739 T^{4} - 185 p T^{5} - 6 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 - T - 36 T^{2} + 73 T^{3} + 1259 T^{4} + 73 p T^{5} - 36 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 47 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 53 | $C_4\times C_2$ | \( 1 - 6 T - 17 T^{2} + 420 T^{3} - 1619 T^{4} + 420 p T^{5} - 17 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 + 3 T - 50 T^{2} - 327 T^{3} + 1969 T^{4} - 327 p T^{5} - 50 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_4\times C_2$ | \( 1 + 4 T - 45 T^{2} - 424 T^{3} + 1049 T^{4} - 424 p T^{5} - 45 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 71 | $C_4\times C_2$ | \( 1 + 15 T + 154 T^{2} + 1245 T^{3} + 7741 T^{4} + 1245 p T^{5} + 154 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 + 4 T - 57 T^{2} - 520 T^{3} + 2081 T^{4} - 520 p T^{5} - 57 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 - 2 T - 75 T^{2} + 308 T^{3} + 5309 T^{4} + 308 p T^{5} - 75 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_4\times C_2$ | \( 1 - 6 T - 47 T^{2} + 780 T^{3} - 779 T^{4} + 780 p T^{5} - 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 97 | $C_4\times C_2$ | \( 1 - 7 T - 48 T^{2} + 1015 T^{3} - 2449 T^{4} + 1015 p T^{5} - 48 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.82893649174927800200913964437, −7.42396389834517857343184190716, −7.38957450172641131166273749011, −7.35282880578821108307230286617, −7.33581250524104030925515002168, −6.56518263730742926592386490519, −6.49094742748738134341241732196, −6.01105138310948492920737718366, −5.80503758436455896158296472628, −5.69060491927036264727887658014, −5.61727120668016419310224964388, −5.25856857005740610670451872142, −5.12014718249656613689065079337, −4.42756399396769183806085103039, −4.38964446451887250315531541308, −4.18844930802716877080359787905, −4.00517394530586940089465546418, −3.32869625136823597935180608635, −3.08118490892394758431057700956, −2.66317511789158502783548007910, −2.35128542778674595783181258781, −2.04171192493874822520990477908, −1.58062411332497186561083618198, −1.15945484992228854286561639339, −0.75657274691484477852318630432,
0.75657274691484477852318630432, 1.15945484992228854286561639339, 1.58062411332497186561083618198, 2.04171192493874822520990477908, 2.35128542778674595783181258781, 2.66317511789158502783548007910, 3.08118490892394758431057700956, 3.32869625136823597935180608635, 4.00517394530586940089465546418, 4.18844930802716877080359787905, 4.38964446451887250315531541308, 4.42756399396769183806085103039, 5.12014718249656613689065079337, 5.25856857005740610670451872142, 5.61727120668016419310224964388, 5.69060491927036264727887658014, 5.80503758436455896158296472628, 6.01105138310948492920737718366, 6.49094742748738134341241732196, 6.56518263730742926592386490519, 7.33581250524104030925515002168, 7.35282880578821108307230286617, 7.38957450172641131166273749011, 7.42396389834517857343184190716, 7.82893649174927800200913964437
