L(s) = 1 | − 4·7-s − 24·17-s + 12·23-s − 24·41-s − 12·47-s + 8·49-s + 32·73-s − 96·79-s − 8·81-s + 32·97-s + 28·103-s + 96·119-s − 48·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 48·161-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
L(s) = 1 | − 1.51·7-s − 5.82·17-s + 2.50·23-s − 3.74·41-s − 1.75·47-s + 8/7·49-s + 3.74·73-s − 10.8·79-s − 8/9·81-s + 3.24·97-s + 2.75·103-s + 8.80·119-s − 4.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 3.78·161-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1375113979\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1375113979\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 8 T^{4} + 94 T^{8} + 8 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( ( 1 + 2 T + 2 T^{2} - 6 T^{3} - 82 T^{4} - 6 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 24 T^{2} + 302 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 62 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 2 T + p T^{2} )^{4}( 1 + 8 T + p T^{2} )^{4} \) |
| 19 | \( ( 1 - 8 T + p T^{2} )^{4}( 1 + 8 T + p T^{2} )^{4} \) |
| 23 | \( ( 1 - 6 T + 18 T^{2} - 102 T^{3} + 542 T^{4} - 102 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 64 T^{2} + 2190 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 2338 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 6 T + 70 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 + 2632 T^{4} + 3107358 T^{8} + 2632 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 6 T + 18 T^{2} + 246 T^{3} + 3326 T^{4} + 246 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 2846 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 112 T^{2} + 9822 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 56 T^{4} - 39549474 T^{8} - 56 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 80 T^{2} + 4878 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 16 T + 128 T^{2} - 1008 T^{3} + 7838 T^{4} - 1008 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 12 T + p T^{2} )^{8} \) |
| 83 | \( 1 + 8200 T^{4} + 98353758 T^{8} + 8200 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 116 T^{2} + 7110 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 16 T + 128 T^{2} - 1392 T^{3} + 15038 T^{4} - 1392 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.87624389375403587567356827819, −3.76615661926936189899057247371, −3.75958787950139593079935773786, −3.63540905327814781097094432246, −3.58583514819973914655508820997, −3.45812846097328755594372608528, −3.04157483676854178671707331919, −2.93479617233850474577153401865, −2.84303004610919970746286465905, −2.80407301239109361826728947618, −2.76925212149475013949940739682, −2.70648893901684646925451224737, −2.51290892666945758661264647190, −2.33607276951438416255648687226, −1.89463147961070180141042112240, −1.85189446431274455332675519526, −1.82054686632295932819624857814, −1.79408366305399868455188577853, −1.72343045919047928345478335549, −1.15880332474627391846915092695, −1.11986585531475726619217518828, −1.01126398101893894452848473263, −0.37872377069726811755973835644, −0.30064109495603448985328720756, −0.090986966732277607534472287152,
0.090986966732277607534472287152, 0.30064109495603448985328720756, 0.37872377069726811755973835644, 1.01126398101893894452848473263, 1.11986585531475726619217518828, 1.15880332474627391846915092695, 1.72343045919047928345478335549, 1.79408366305399868455188577853, 1.82054686632295932819624857814, 1.85189446431274455332675519526, 1.89463147961070180141042112240, 2.33607276951438416255648687226, 2.51290892666945758661264647190, 2.70648893901684646925451224737, 2.76925212149475013949940739682, 2.80407301239109361826728947618, 2.84303004610919970746286465905, 2.93479617233850474577153401865, 3.04157483676854178671707331919, 3.45812846097328755594372608528, 3.58583514819973914655508820997, 3.63540905327814781097094432246, 3.75958787950139593079935773786, 3.76615661926936189899057247371, 3.87624389375403587567356827819
Plot not available for L-functions of degree greater than 10.