| L(s) = 1 | + 2-s − 2·3-s + 6·5-s − 2·6-s − 3·7-s − 5·8-s + 9-s + 6·10-s − 4·11-s − 3·14-s − 12·15-s − 3·16-s − 17-s + 18-s + 6·19-s + 6·21-s − 4·22-s − 4·23-s + 10·24-s + 11·25-s + 2·27-s − 29-s − 12·30-s − 2·31-s + 8·33-s − 34-s − 18·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 2.68·5-s − 0.816·6-s − 1.13·7-s − 1.76·8-s + 1/3·9-s + 1.89·10-s − 1.20·11-s − 0.801·14-s − 3.09·15-s − 3/4·16-s − 0.242·17-s + 0.235·18-s + 1.37·19-s + 1.30·21-s − 0.852·22-s − 0.834·23-s + 2.04·24-s + 11/5·25-s + 0.384·27-s − 0.185·29-s − 2.19·30-s − 0.359·31-s + 1.39·33-s − 0.171·34-s − 3.04·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.105659340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.105659340\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - T + T^{2} + p^{2} T^{3} - 3 p T^{4} + p^{3} T^{5} + p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 + 3 T - 3 T^{2} - 6 T^{3} + 32 T^{4} - 6 p T^{5} - 3 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + T - 29 T^{2} - 4 T^{3} + 594 T^{4} - 4 p T^{5} - 29 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 6 T + 6 T^{2} + 48 T^{3} - 145 T^{4} + 48 p T^{5} + 6 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + T - 19 T^{2} - 38 T^{3} - 470 T^{4} - 38 p T^{5} - 19 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + T + 58 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 11 T + 21 T^{2} - 286 T^{3} + 4154 T^{4} - 286 p T^{5} + 21 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - T - 77 T^{2} + 4 T^{3} + 4362 T^{4} + 4 p T^{5} - 77 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 5 T - 63 T^{2} + 10 T^{3} + 4820 T^{4} + 10 p T^{5} - 63 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 11 T + 98 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 14 T + 46 T^{2} + 448 T^{3} + 7455 T^{4} + 448 p T^{5} + 46 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 16 T + 87 T^{2} + 752 T^{3} + 9224 T^{4} + 752 p T^{5} + 87 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 5 T - 111 T^{2} - 10 T^{3} + 12332 T^{4} - 10 p T^{5} - 111 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 14 T + 125 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 12 T + 165 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 15 T + 210 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 10 T + 174 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 18 T + 82 T^{2} - 1152 T^{3} + 21807 T^{4} - 1152 p T^{5} + 82 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 13 T - 63 T^{2} + 494 T^{3} + 28022 T^{4} + 494 p T^{5} - 63 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.64651741656644496436944365778, −7.63661169518696744267818112828, −7.61643638444700356476380948937, −6.68061807172844379866689607232, −6.63576926055383295530009825877, −6.54295540324409244938946421240, −6.37901254865646015753273483190, −6.00364067197508502832458875691, −5.93630573018603626252572068759, −5.70555074024199559945781498270, −5.28989216872990357745754169094, −5.25738935679637812614144233208, −5.18508400443900215780810535581, −5.04955947583767624716138085244, −4.23115734252547895190767245394, −4.01362408826395669446187411039, −3.66302251458454301614862436545, −3.45110746090661217016964648624, −2.98623095044222681985502288703, −2.74794410724370095618615346792, −2.38450978439024155979693654599, −2.11054503878639634575854936905, −1.90846095971533647900261052386, −0.908229563814700896464000294106, −0.54296060004910128312835454682,
0.54296060004910128312835454682, 0.908229563814700896464000294106, 1.90846095971533647900261052386, 2.11054503878639634575854936905, 2.38450978439024155979693654599, 2.74794410724370095618615346792, 2.98623095044222681985502288703, 3.45110746090661217016964648624, 3.66302251458454301614862436545, 4.01362408826395669446187411039, 4.23115734252547895190767245394, 5.04955947583767624716138085244, 5.18508400443900215780810535581, 5.25738935679637812614144233208, 5.28989216872990357745754169094, 5.70555074024199559945781498270, 5.93630573018603626252572068759, 6.00364067197508502832458875691, 6.37901254865646015753273483190, 6.54295540324409244938946421240, 6.63576926055383295530009825877, 6.68061807172844379866689607232, 7.61643638444700356476380948937, 7.63661169518696744267818112828, 7.64651741656644496436944365778
