| L(s) = 1 | − 2-s + 4-s + 2·5-s − 2·7-s + 2·8-s − 2·10-s + 2·11-s − 6·13-s + 2·14-s + 8·17-s + 2·20-s − 2·22-s − 6·23-s + 25-s + 6·26-s − 2·28-s + 10·29-s + 4·31-s + 4·32-s − 8·34-s − 4·35-s + 8·37-s + 4·40-s − 2·41-s + 6·43-s + 2·44-s + 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.755·7-s + 0.707·8-s − 0.632·10-s + 0.603·11-s − 1.66·13-s + 0.534·14-s + 1.94·17-s + 0.447·20-s − 0.426·22-s − 1.25·23-s + 1/5·25-s + 1.17·26-s − 0.377·28-s + 1.85·29-s + 0.718·31-s + 0.707·32-s − 1.37·34-s − 0.676·35-s + 1.31·37-s + 0.632·40-s − 0.312·41-s + 0.914·43-s + 0.301·44-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.283198028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.283198028\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + T - 3 T^{3} - 5 T^{4} - 3 p T^{5} + p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^3$ | \( 1 + 2 T + 2 T^{2} - 24 T^{3} - 73 T^{4} - 24 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - 6 T^{2} + 24 T^{3} - 65 T^{4} + 24 p T^{5} - 6 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 6 T + 14 T^{2} - 24 T^{3} - 153 T^{4} - 24 p T^{5} + 14 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 25 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 10 T + 30 T^{2} - 120 T^{3} + 1159 T^{4} - 120 p T^{5} + 30 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 4 T - 37 T^{2} + 36 T^{3} + 1352 T^{4} + 36 p T^{5} - 37 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 + 2 T - 66 T^{2} - 24 T^{3} + 3055 T^{4} - 24 p T^{5} - 66 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 6 T - 46 T^{2} + 24 T^{3} + 3327 T^{4} + 24 p T^{5} - 46 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 4 T - 30 T^{2} - 192 T^{3} - 845 T^{4} - 192 p T^{5} - 30 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 4 T + 97 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 10 T - 30 T^{2} + 120 T^{3} + 7519 T^{4} + 120 p T^{5} - 30 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 6 T - 43 T^{2} - 258 T^{3} + 324 T^{4} - 258 p T^{5} - 43 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 16 T + 110 T^{2} - 192 T^{3} - 325 T^{4} - 192 p T^{5} + 110 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 22 T + 250 T^{2} + 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 18 T + 214 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 16 T + 47 T^{2} - 816 T^{3} + 17216 T^{4} - 816 p T^{5} + 47 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 3 T - 74 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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
−8.095683912958206671642175492552, −7.86594515876090155904818596983, −7.68453296764371841843216192419, −7.60040465057435819924002198578, −7.32436297571220941291849413240, −6.74348904683304637141656124884, −6.64433614507061457471715599178, −6.41305659278662071023006572924, −6.40157761051779049411797018760, −5.87912308039798783819519929362, −5.59092157352171441693458352360, −5.49255452988697611010933655776, −5.04327687502710730802665851930, −4.87423338705804572557386282898, −4.45468063577354559827979660202, −4.12983989276700198243143950692, −4.05210494209291784399150028066, −3.32602525243438630450987156500, −3.18423954930775950548306505036, −2.86433714449689844625733203900, −2.36316199488495065725759961756, −2.21165622927383267628509703333, −1.65157750915870385025065854640, −1.09931233469410282585184932909, −0.76611173884785771570507339475,
0.76611173884785771570507339475, 1.09931233469410282585184932909, 1.65157750915870385025065854640, 2.21165622927383267628509703333, 2.36316199488495065725759961756, 2.86433714449689844625733203900, 3.18423954930775950548306505036, 3.32602525243438630450987156500, 4.05210494209291784399150028066, 4.12983989276700198243143950692, 4.45468063577354559827979660202, 4.87423338705804572557386282898, 5.04327687502710730802665851930, 5.49255452988697611010933655776, 5.59092157352171441693458352360, 5.87912308039798783819519929362, 6.40157761051779049411797018760, 6.41305659278662071023006572924, 6.64433614507061457471715599178, 6.74348904683304637141656124884, 7.32436297571220941291849413240, 7.60040465057435819924002198578, 7.68453296764371841843216192419, 7.86594515876090155904818596983, 8.095683912958206671642175492552
