| L(s) = 1 | + 2-s − 2·3-s + 2·4-s − 5-s − 2·6-s − 2·7-s + 3·9-s − 10-s − 4·12-s + 13-s − 2·14-s + 2·15-s − 5·17-s + 3·18-s + 6·19-s − 2·20-s + 4·21-s + 8·23-s + 5·25-s + 26-s − 4·28-s + 9·29-s + 2·30-s + 2·31-s − 11·32-s − 5·34-s + 2·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 4-s − 0.447·5-s − 0.816·6-s − 0.755·7-s + 9-s − 0.316·10-s − 1.15·12-s + 0.277·13-s − 0.534·14-s + 0.516·15-s − 1.21·17-s + 0.707·18-s + 1.37·19-s − 0.447·20-s + 0.872·21-s + 1.66·23-s + 25-s + 0.196·26-s − 0.755·28-s + 1.67·29-s + 0.365·30-s + 0.359·31-s − 1.94·32-s − 0.857·34-s + 0.338·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.081561645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.081561645\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | | \( 1 \) |
| good | 2 | $C_4\times C_2$ | \( 1 - T - T^{2} + 3 T^{3} - T^{4} + 3 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | $C_4\times C_2$ | \( 1 + 2 T + T^{2} - 4 T^{3} - 11 T^{4} - 4 p T^{5} + p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_4\times C_2$ | \( 1 + T - 4 T^{2} - 9 T^{3} + 11 T^{4} - 9 p T^{5} - 4 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 + 2 T - 3 T^{2} - 20 T^{3} - 19 T^{4} - 20 p T^{5} - 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_4\times C_2$ | \( 1 - T - 12 T^{2} + 25 T^{3} + 131 T^{4} + 25 p T^{5} - 12 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 + 5 T + 8 T^{2} - 45 T^{3} - 361 T^{4} - 45 p T^{5} + 8 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 - 6 T + 17 T^{2} + 12 T^{3} - 395 T^{4} + 12 p T^{5} + 17 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 29 | $C_4\times C_2$ | \( 1 - 9 T + 52 T^{2} - 207 T^{3} + 355 T^{4} - 207 p T^{5} + 52 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 - 2 T - 27 T^{2} + 116 T^{3} + 605 T^{4} + 116 p T^{5} - 27 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 - 3 T - 28 T^{2} + 195 T^{3} + 451 T^{4} + 195 p T^{5} - 28 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_4\times C_2$ | \( 1 + 5 T - 16 T^{2} - 285 T^{3} - 769 T^{4} - 285 p T^{5} - 16 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 47 | $C_4\times C_2$ | \( 1 + 2 T - 43 T^{2} - 180 T^{3} + 1661 T^{4} - 180 p T^{5} - 43 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_4\times C_2$ | \( 1 + 9 T + 28 T^{2} - 225 T^{3} - 3509 T^{4} - 225 p T^{5} + 28 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 + 8 T + 5 T^{2} - 432 T^{3} - 3751 T^{4} - 432 p T^{5} + 5 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_4\times C_2$ | \( 1 - 6 T - 25 T^{2} + 516 T^{3} - 1571 T^{4} + 516 p T^{5} - 25 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 71 | $C_4\times C_2$ | \( 1 + 12 T + 73 T^{2} + 24 T^{3} - 4895 T^{4} + 24 p T^{5} + 73 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 + 2 T - 69 T^{2} - 284 T^{3} + 4469 T^{4} - 284 p T^{5} - 69 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 + 10 T + 21 T^{2} - 580 T^{3} - 7459 T^{4} - 580 p T^{5} + 21 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_4\times C_2$ | \( 1 - 6 T - 47 T^{2} + 780 T^{3} - 779 T^{4} + 780 p T^{5} - 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 97 | $C_4\times C_2$ | \( 1 - 13 T + 72 T^{2} + 325 T^{3} - 11209 T^{4} + 325 p T^{5} + 72 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−10.01513309344965017522234829101, −9.859451708039210727580304439824, −9.287048105505832980344628472181, −9.166248815155562469040968109968, −8.854029960385424933647284807661, −8.436618857388649544125903002040, −8.414667415648191824996173831256, −7.75384528321022666444419855021, −7.35947481729673743681244868233, −7.07909311579771284569498995016, −7.01784743536703267942058993345, −6.64579030538668847188487540763, −6.27867351960954730284903549872, −6.24028125433049972033214549607, −5.70061139574055580956689523721, −5.36565454275731227588423987372, −4.99591922705429192385995306899, −4.82177005491798371864729318047, −4.27241533587434742940651276809, −4.06639688122683709048418171842, −3.19636470591864378394209702976, −3.01375699281306565671805504984, −2.95322586359254882503304593617, −1.92659410493947734409567211885, −0.995596748791839602684278442338,
0.995596748791839602684278442338, 1.92659410493947734409567211885, 2.95322586359254882503304593617, 3.01375699281306565671805504984, 3.19636470591864378394209702976, 4.06639688122683709048418171842, 4.27241533587434742940651276809, 4.82177005491798371864729318047, 4.99591922705429192385995306899, 5.36565454275731227588423987372, 5.70061139574055580956689523721, 6.24028125433049972033214549607, 6.27867351960954730284903549872, 6.64579030538668847188487540763, 7.01784743536703267942058993345, 7.07909311579771284569498995016, 7.35947481729673743681244868233, 7.75384528321022666444419855021, 8.414667415648191824996173831256, 8.436618857388649544125903002040, 8.854029960385424933647284807661, 9.166248815155562469040968109968, 9.287048105505832980344628472181, 9.859451708039210727580304439824, 10.01513309344965017522234829101
