| L(s) = 1 | + 4·3-s + 3·7-s + 10·9-s + 5·11-s + 4·13-s + 7·17-s + 3·19-s + 12·21-s + 8·23-s + 20·27-s + 2·29-s + 5·31-s + 20·33-s − 37-s + 16·39-s + 7·41-s + 6·43-s − 8·49-s + 28·51-s + 6·53-s + 12·57-s − 9·59-s + 19·61-s + 30·63-s − 4·67-s + 32·69-s + 19·71-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1.13·7-s + 10/3·9-s + 1.50·11-s + 1.10·13-s + 1.69·17-s + 0.688·19-s + 2.61·21-s + 1.66·23-s + 3.84·27-s + 0.371·29-s + 0.898·31-s + 3.48·33-s − 0.164·37-s + 2.56·39-s + 1.09·41-s + 0.914·43-s − 8/7·49-s + 3.92·51-s + 0.824·53-s + 1.58·57-s − 1.17·59-s + 2.43·61-s + 3.77·63-s − 0.488·67-s + 3.85·69-s + 2.25·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(77.16001919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(77.16001919\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 7 | $C_2 \wr S_4$ | \( 1 - 3 T + 17 T^{2} - 24 T^{3} + 120 T^{4} - 24 p T^{5} + 17 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 5 T + 15 T^{2} - 6 T^{3} + 16 T^{4} - 6 p T^{5} + 15 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 7 T + 39 T^{2} - 186 T^{3} + 964 T^{4} - 186 p T^{5} + 39 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 3 T + 34 T^{2} - 27 T^{3} + 606 T^{4} - 27 p T^{5} + 34 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 8 T + 84 T^{2} - 420 T^{3} + 2614 T^{4} - 420 p T^{5} + 84 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 2 T + 60 T^{2} + 36 T^{3} + 1741 T^{4} + 36 p T^{5} + 60 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 5 T + 79 T^{2} - 158 T^{3} + 2548 T^{4} - 158 p T^{5} + 79 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + T + 2 p T^{2} + 123 T^{3} + 3374 T^{4} + 123 p T^{5} + 2 p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 7 T + 72 T^{2} + 99 T^{3} + 298 T^{4} + 99 p T^{5} + 72 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 6 T + 80 T^{2} - 714 T^{3} + 3174 T^{4} - 714 p T^{5} + 80 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $S_4\times C_2$ | \( 1 + 60 T^{2} + 18 T^{3} + 5171 T^{4} + 18 p T^{5} + 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 6 T - 12 T^{2} + 24 T^{3} + 3257 T^{4} + 24 p T^{5} - 12 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 9 T + 107 T^{2} + 576 T^{3} + 6690 T^{4} + 576 p T^{5} + 107 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 19 T + 319 T^{2} - 3286 T^{3} + 30796 T^{4} - 3286 p T^{5} + 319 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 4 T + 142 T^{2} + 382 T^{3} + 12817 T^{4} + 382 p T^{5} + 142 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 19 T + 374 T^{2} - 3967 T^{3} + 42262 T^{4} - 3967 p T^{5} + 374 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 6 T + 16 T^{2} + 486 T^{3} + 11694 T^{4} + 486 p T^{5} + 16 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 9 T + 272 T^{2} + 1893 T^{3} + 30882 T^{4} + 1893 p T^{5} + 272 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 21 T + 483 T^{2} - 5676 T^{3} + 66866 T^{4} - 5676 p T^{5} + 483 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 14 T + 276 T^{2} - 3402 T^{3} + 34726 T^{4} - 3402 p T^{5} + 276 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 6 T + 344 T^{2} + 1434 T^{3} + 47598 T^{4} + 1434 p T^{5} + 344 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.50347107945682503032333749406, −5.10398295918201633320956077844, −5.06118766896553447822183126211, −5.04110235529396239488357133282, −4.77754857782582545401027721121, −4.41920179953936698240972233904, −4.35592543162041928351219042786, −4.03366154887034860787225651138, −4.03073710436444799073038821635, −3.80291502276183168449723076076, −3.55415638166617779235034034498, −3.26982534816465364611938533866, −3.22923027695169606678035572960, −3.18544817190239063444438850941, −2.81295908393322705707309386519, −2.57486759573940413777479287480, −2.54208834624490875512241101630, −1.98638976906454830002208137368, −1.81654394941875427426578183329, −1.72627625727315830890220328400, −1.66216270383233915293556715563, −1.02869302101362604352853902512, −0.863566553360371787891523876390, −0.862649054206748742484748454290, −0.76451537744967454698853416196,
0.76451537744967454698853416196, 0.862649054206748742484748454290, 0.863566553360371787891523876390, 1.02869302101362604352853902512, 1.66216270383233915293556715563, 1.72627625727315830890220328400, 1.81654394941875427426578183329, 1.98638976906454830002208137368, 2.54208834624490875512241101630, 2.57486759573940413777479287480, 2.81295908393322705707309386519, 3.18544817190239063444438850941, 3.22923027695169606678035572960, 3.26982534816465364611938533866, 3.55415638166617779235034034498, 3.80291502276183168449723076076, 4.03073710436444799073038821635, 4.03366154887034860787225651138, 4.35592543162041928351219042786, 4.41920179953936698240972233904, 4.77754857782582545401027721121, 5.04110235529396239488357133282, 5.06118766896553447822183126211, 5.10398295918201633320956077844, 5.50347107945682503032333749406
