| L(s) = 1 | + 4·3-s + 4·5-s − 2·7-s + 10·9-s − 4·11-s − 4·13-s + 16·15-s + 4·17-s − 2·19-s − 8·21-s + 8·23-s + 4·25-s + 20·27-s + 6·29-s + 8·31-s − 16·33-s − 8·35-s + 14·37-s − 16·39-s + 10·41-s − 12·43-s + 40·45-s − 2·47-s − 4·49-s + 16·51-s + 12·53-s − 16·55-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1.78·5-s − 0.755·7-s + 10/3·9-s − 1.20·11-s − 1.10·13-s + 4.13·15-s + 0.970·17-s − 0.458·19-s − 1.74·21-s + 1.66·23-s + 4/5·25-s + 3.84·27-s + 1.11·29-s + 1.43·31-s − 2.78·33-s − 1.35·35-s + 2.30·37-s − 2.56·39-s + 1.56·41-s − 1.82·43-s + 5.96·45-s − 0.291·47-s − 4/7·49-s + 2.24·51-s + 1.64·53-s − 2.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(44.23945777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(44.23945777\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{4} \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 - 4 T + 12 T^{2} - 34 T^{3} + 98 T^{4} - 34 p T^{5} + 12 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 2 T + 8 T^{2} + 38 T^{3} + 78 T^{4} + 38 p T^{5} + 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 4 T + 54 T^{2} - 198 T^{3} + 1262 T^{4} - 198 p T^{5} + 54 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 2 T + 56 T^{2} + 110 T^{3} + 1470 T^{4} + 110 p T^{5} + 56 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 8 T + 104 T^{2} - 540 T^{3} + 3710 T^{4} - 540 p T^{5} + 104 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 6 T + 82 T^{2} - 456 T^{3} + 3046 T^{4} - 456 p T^{5} + 82 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 8 T + 72 T^{2} - 370 T^{3} + 2766 T^{4} - 370 p T^{5} + 72 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 14 T + 116 T^{2} - 994 T^{3} + 7414 T^{4} - 994 p T^{5} + 116 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 10 T + 148 T^{2} - 1034 T^{3} + 9166 T^{4} - 1034 p T^{5} + 148 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 12 T + 206 T^{2} + 1566 T^{3} + 14002 T^{4} + 1566 p T^{5} + 206 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 2 T + 96 T^{2} + 498 T^{3} + 4958 T^{4} + 498 p T^{5} + 96 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 12 T + 116 T^{2} - 724 T^{3} + 6182 T^{4} - 724 p T^{5} + 116 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 8 T + 60 T^{2} + 520 T^{3} - 62 p T^{4} + 520 p T^{5} + 60 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 10 T + 188 T^{2} + 1734 T^{3} + 15942 T^{4} + 1734 p T^{5} + 188 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 8 T + 148 T^{2} - 1162 T^{3} + 14302 T^{4} - 1162 p T^{5} + 148 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 26 T + 432 T^{2} - 5258 T^{3} + 51230 T^{4} - 5258 p T^{5} + 432 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 14 T + 256 T^{2} - 2422 T^{3} + 25414 T^{4} - 2422 p T^{5} + 256 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 16 T + 258 T^{2} - 2594 T^{3} + 29178 T^{4} - 2594 p T^{5} + 258 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 8 T + 228 T^{2} + 2200 T^{3} + 24006 T^{4} + 2200 p T^{5} + 228 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 18 T + 432 T^{2} - 4604 T^{3} + 59690 T^{4} - 4604 p T^{5} + 432 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 14 T + 268 T^{2} - 2842 T^{3} + 37558 T^{4} - 2842 p T^{5} + 268 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 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
−5.64819766113535223615983155680, −5.18378589719848449259635802950, −5.17829146916666207896280675844, −5.07010393053866752159800843650, −5.01397126569040178781438642492, −4.53411514219471754818187551590, −4.38387996097988117187839600586, −4.28695946280212489853299360232, −4.26116605857346343492384544411, −3.63147639661787255190637294927, −3.44804604564974339517074778603, −3.40900975624857467904685232707, −3.27718830145393997881349567772, −2.93579140281788973807250441021, −2.79587150851093110671750453355, −2.64149205265954478912518909603, −2.46983642489964295475084776557, −2.16343107819104626980351016239, −1.96559729497285989929488315774, −1.91608356732342431965004334102, −1.86459995994052622738307378980, −1.08740082906941530998716811507, −0.77102126455346348394857305366, −0.75030017997666982976146753102, −0.66663913647106057803363367263,
0.66663913647106057803363367263, 0.75030017997666982976146753102, 0.77102126455346348394857305366, 1.08740082906941530998716811507, 1.86459995994052622738307378980, 1.91608356732342431965004334102, 1.96559729497285989929488315774, 2.16343107819104626980351016239, 2.46983642489964295475084776557, 2.64149205265954478912518909603, 2.79587150851093110671750453355, 2.93579140281788973807250441021, 3.27718830145393997881349567772, 3.40900975624857467904685232707, 3.44804604564974339517074778603, 3.63147639661787255190637294927, 4.26116605857346343492384544411, 4.28695946280212489853299360232, 4.38387996097988117187839600586, 4.53411514219471754818187551590, 5.01397126569040178781438642492, 5.07010393053866752159800843650, 5.17829146916666207896280675844, 5.18378589719848449259635802950, 5.64819766113535223615983155680
