| L(s) = 1 | − 4·2-s + 3-s + 12·4-s + 2·5-s − 4·6-s − 7-s − 25·8-s − 8·10-s + 12·12-s + 3·13-s + 4·14-s + 2·15-s + 45·16-s + 3·17-s − 5·19-s + 24·20-s − 21-s − 8·23-s − 25·24-s + 5·25-s − 12·26-s − 12·28-s − 6·29-s − 8·30-s + 13·31-s − 66·32-s − 12·34-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 0.577·3-s + 6·4-s + 0.894·5-s − 1.63·6-s − 0.377·7-s − 8.83·8-s − 2.52·10-s + 3.46·12-s + 0.832·13-s + 1.06·14-s + 0.516·15-s + 45/4·16-s + 0.727·17-s − 1.14·19-s + 5.36·20-s − 0.218·21-s − 1.66·23-s − 5.10·24-s + 25-s − 2.35·26-s − 2.26·28-s − 1.11·29-s − 1.46·30-s + 2.33·31-s − 11.6·32-s − 2.05·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.041644855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.041644855\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_4\times C_2$ | \( 1 + p^{2} T + p^{2} T^{2} - 7 T^{3} - 21 T^{4} - 7 p T^{5} + p^{4} T^{6} + p^{5} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2:C_4$ | \( 1 - 2 T - T^{2} - 8 T^{3} + 41 T^{4} - 8 p T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 + T - 6 T^{2} - 13 T^{3} + 29 T^{4} - 13 p T^{5} - 6 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 3 T + 6 T^{2} - 59 T^{3} + 339 T^{4} - 59 p T^{5} + 6 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 3 T + 37 T^{2} - 45 T^{3} + 676 T^{4} - 45 p T^{5} + 37 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 5 T + 21 T^{2} + 145 T^{3} + 956 T^{4} + 145 p T^{5} + 21 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_4\times C_2$ | \( 1 + 6 T + 7 T^{2} - 132 T^{3} - 995 T^{4} - 132 p T^{5} + 7 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 13 T + 63 T^{2} - 11 p T^{3} + 80 p T^{4} - 11 p^{2} T^{5} + 63 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + T - 6 T^{2} + 197 T^{3} + 1379 T^{4} + 197 p T^{5} - 6 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 7 T - 22 T^{2} - 161 T^{3} + 575 T^{4} - 161 p T^{5} - 22 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 41 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 8 T + 67 T^{2} + 610 T^{3} + 6231 T^{4} + 610 p T^{5} + 67 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + T - 47 T^{2} + 155 T^{3} + 2916 T^{4} + 155 p T^{5} - 47 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 11 T + 17 T^{2} - 577 T^{3} - 6700 T^{4} - 577 p T^{5} + 17 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 24 T + 245 T^{2} + 1896 T^{3} + 15169 T^{4} + 1896 p T^{5} + 245 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 3 T + 125 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 20 T + 119 T^{2} + 470 T^{3} + 4871 T^{4} + 470 p T^{5} + 119 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 16 T + 63 T^{2} - 470 T^{3} + 8141 T^{4} - 470 p T^{5} + 63 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 - 11 T + 42 T^{2} + 407 T^{3} - 7795 T^{4} + 407 p T^{5} + 42 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 7 T - 14 T^{2} - 641 T^{3} + 11409 T^{4} - 641 p T^{5} - 14 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 12 T + 209 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 12 T - 43 T^{2} - 900 T^{3} - 1859 T^{4} - 900 p T^{5} - 43 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.314039392246169966491289403067, −8.222513574767645156586956069345, −7.85411342139780790688105541150, −7.63068183853984660780592469763, −7.56054065528193864771209263560, −7.18073510245233359133889444389, −6.67904994335795583275622616156, −6.67875773645983708311433837772, −6.60439520670785799557477451536, −6.12797290465361444543822878569, −5.97071759457729332950087177658, −5.73836767489756991819298644282, −5.52201428620164726692382024267, −4.78225054740279230987821264256, −4.65307074154301162117954604606, −4.12845160631002208742763406072, −3.87116854785734742989735272720, −3.09946023848881013052337308676, −3.04794012998530882308908904002, −2.90379195700167494169822798263, −2.26750602750295708225563223956, −1.85592026815003287843517177974, −1.63890602221627513228706262064, −1.51156156693103737334209707350, −0.61893711627629929726786056548,
0.61893711627629929726786056548, 1.51156156693103737334209707350, 1.63890602221627513228706262064, 1.85592026815003287843517177974, 2.26750602750295708225563223956, 2.90379195700167494169822798263, 3.04794012998530882308908904002, 3.09946023848881013052337308676, 3.87116854785734742989735272720, 4.12845160631002208742763406072, 4.65307074154301162117954604606, 4.78225054740279230987821264256, 5.52201428620164726692382024267, 5.73836767489756991819298644282, 5.97071759457729332950087177658, 6.12797290465361444543822878569, 6.60439520670785799557477451536, 6.67875773645983708311433837772, 6.67904994335795583275622616156, 7.18073510245233359133889444389, 7.56054065528193864771209263560, 7.63068183853984660780592469763, 7.85411342139780790688105541150, 8.222513574767645156586956069345, 8.314039392246169966491289403067
