| L(s) = 1 | − 2·2-s + 3·4-s + 2·5-s − 2·7-s − 2·8-s − 4·10-s + 2·11-s + 8·13-s + 4·14-s − 8·17-s + 2·19-s + 6·20-s − 4·22-s − 14·23-s + 11·25-s − 16·26-s − 6·28-s + 32·29-s + 4·31-s + 6·32-s + 16·34-s − 4·35-s − 4·37-s − 4·38-s − 4·40-s + 24·41-s + 4·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.894·5-s − 0.755·7-s − 0.707·8-s − 1.26·10-s + 0.603·11-s + 2.21·13-s + 1.06·14-s − 1.94·17-s + 0.458·19-s + 1.34·20-s − 0.852·22-s − 2.91·23-s + 11/5·25-s − 3.13·26-s − 1.13·28-s + 5.94·29-s + 0.718·31-s + 1.06·32-s + 2.74·34-s − 0.676·35-s − 0.657·37-s − 0.648·38-s − 0.632·40-s + 3.74·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.076455884\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.076455884\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p T + T^{2} - p T^{3} - 3 T^{4} - p^{2} T^{5} + p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - 17 T^{2} + 2 T^{3} + 276 T^{4} + 2 p T^{5} - 17 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 14 T + 103 T^{2} + 658 T^{3} + 3612 T^{4} + 658 p T^{5} + 103 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 16 T + 120 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 4 T - 32 T^{2} + 56 T^{3} + 847 T^{4} + 56 p T^{5} - 32 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T - 12 T^{2} - 184 T^{3} - 1177 T^{4} - 184 p T^{5} - 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 2 T + 37 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 6 T - 65 T^{2} + 42 T^{3} + 6300 T^{4} + 42 p T^{5} - 65 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 88 T^{2} + 4935 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 8 T - 62 T^{2} - 64 T^{3} + 8619 T^{4} - 64 p T^{5} - 62 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 18 T + 129 T^{2} + 1314 T^{3} + 14540 T^{4} + 1314 p T^{5} + 129 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 8 T - 78 T^{2} + 64 T^{3} + 11387 T^{4} + 64 p T^{5} - 78 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 20 T + 224 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 2 T + 57 T^{2} - 398 T^{3} - 2812 T^{4} - 398 p T^{5} + 57 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^3$ | \( 1 - 156 T^{2} + 18095 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 10 T + 173 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 20 T + 124 T^{2} + 1960 T^{3} + 32655 T^{4} + 1960 p T^{5} + 124 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 12 T + 222 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.94731530880997496189134532735, −6.65304394427374543884662817988, −6.56825690732477346074216675820, −6.36491397990409890641852420729, −6.12773734901224289190276978372, −6.06080831543742067929405791721, −5.93635312021976388589318394900, −5.70412713209926256688911791401, −5.19283218022745620949980659838, −4.73113598487003842251774214443, −4.59440653043144433258137485094, −4.52784593120571077127466866975, −4.34530716320002866362518025657, −3.94767333702331543073631775269, −3.62429219525596913081534353430, −3.47367316644034343958887515853, −2.78274409209722813728887381340, −2.67935118240829758646092652952, −2.67838375556738888132986943878, −2.37913498291999848905532927465, −1.91603524934336982325275697130, −1.32670646238108388717324112586, −1.24277589926343882162253119135, −0.964728416692286553152620430796, −0.55013345329409728298202543782,
0.55013345329409728298202543782, 0.964728416692286553152620430796, 1.24277589926343882162253119135, 1.32670646238108388717324112586, 1.91603524934336982325275697130, 2.37913498291999848905532927465, 2.67838375556738888132986943878, 2.67935118240829758646092652952, 2.78274409209722813728887381340, 3.47367316644034343958887515853, 3.62429219525596913081534353430, 3.94767333702331543073631775269, 4.34530716320002866362518025657, 4.52784593120571077127466866975, 4.59440653043144433258137485094, 4.73113598487003842251774214443, 5.19283218022745620949980659838, 5.70412713209926256688911791401, 5.93635312021976388589318394900, 6.06080831543742067929405791721, 6.12773734901224289190276978372, 6.36491397990409890641852420729, 6.56825690732477346074216675820, 6.65304394427374543884662817988, 6.94731530880997496189134532735
