| L(s) = 1 | − 2-s + 4-s − 4·5-s + 4·7-s − 2·8-s + 4·10-s + 4·11-s + 2·13-s − 4·14-s + 2·16-s + 4·19-s − 4·20-s − 4·22-s + 10·25-s − 2·26-s + 4·28-s + 10·29-s + 6·31-s − 4·32-s − 16·35-s + 10·37-s − 4·38-s + 8·40-s + 2·41-s + 18·43-s + 4·44-s − 18·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.78·5-s + 1.51·7-s − 0.707·8-s + 1.26·10-s + 1.20·11-s + 0.554·13-s − 1.06·14-s + 1/2·16-s + 0.917·19-s − 0.894·20-s − 0.852·22-s + 2·25-s − 0.392·26-s + 0.755·28-s + 1.85·29-s + 1.07·31-s − 0.707·32-s − 2.70·35-s + 1.64·37-s − 0.648·38-s + 1.26·40-s + 0.312·41-s + 2.74·43-s + 0.603·44-s − 2.62·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.184764897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.184764897\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 2 | $C_2^3:S_4$ | \( 1 + T + T^{3} + T^{4} + p T^{5} + p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 4 T + 31 T^{2} - 114 T^{3} + 476 T^{4} - 114 p T^{5} + 31 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 2 T + 23 T^{2} - 2 T^{3} + 212 T^{4} - 2 p T^{5} + 23 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 23 T^{2} - 4 T^{3} + 576 T^{4} - 4 p T^{5} + 23 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 4 T + 44 T^{2} - 212 T^{3} + 1089 T^{4} - 212 p T^{5} + 44 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 47 T^{2} - 4 T^{3} + 1476 T^{4} - 4 p T^{5} + 47 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 10 T + 88 T^{2} - 614 T^{3} + 3806 T^{4} - 614 p T^{5} + 88 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 6 T + 85 T^{2} - 522 T^{3} + 3368 T^{4} - 522 p T^{5} + 85 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 10 T + 152 T^{2} - 998 T^{3} + 8478 T^{4} - 998 p T^{5} + 152 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 2 T + 81 T^{2} + 90 T^{3} + 2956 T^{4} + 90 p T^{5} + 81 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 18 T + 241 T^{2} - 54 p T^{3} + 16832 T^{4} - 54 p^{2} T^{5} + 241 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 18 T + 189 T^{2} + 1426 T^{3} + 9460 T^{4} + 1426 p T^{5} + 189 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 4 T + 128 T^{2} - 628 T^{3} + 8061 T^{4} - 628 p T^{5} + 128 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 2 T + 204 T^{2} - 314 T^{3} + 17254 T^{4} - 314 p T^{5} + 204 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 8 T + 223 T^{2} - 1312 T^{3} + 19964 T^{4} - 1312 p T^{5} + 223 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 10 T + 169 T^{2} - 1814 T^{3} + 14168 T^{4} - 1814 p T^{5} + 169 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 8 T + 4 T^{2} + 504 T^{3} - 3034 T^{4} + 504 p T^{5} + 4 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 10 T + 299 T^{2} + 2054 T^{3} + 32712 T^{4} + 2054 p T^{5} + 299 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 12 T + 243 T^{2} - 2092 T^{3} + 27696 T^{4} - 2092 p T^{5} + 243 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 24 T + 426 T^{2} + 5008 T^{3} + 51355 T^{4} + 5008 p T^{5} + 426 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 8 T + 141 T^{2} - 1338 T^{3} + 14260 T^{4} - 1338 p T^{5} + 141 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 10 T + 160 T^{2} - 814 T^{3} + 16542 T^{4} - 814 p T^{5} + 160 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.10446625739659078882182656826, −7.01788168277708826106803488149, −7.01504509627684399848489245223, −6.49504209158974204972098514929, −6.38673286049943538830405322446, −6.27201680063656869747919254524, −5.71047682058079076764186071235, −5.62290696045913103994483902539, −5.57586001860482077141586743162, −4.87168406933367139638899198053, −4.74945753477362831722670801586, −4.70527548286844812828528285987, −4.36173573418078513277175123221, −4.13712067953685228773405927587, −3.80888077538634852637200363057, −3.73475127956385771274174590578, −3.18750182489063981503494188642, −2.96949962833681951052352765362, −2.95874398382912614385956150552, −2.23701096439255923637237990751, −2.13188469695963530738113665950, −1.64982597012910951397424024223, −1.01672450645728970984382158112, −0.794588120318086899729737301335, −0.792626379523688210897945799582,
0.792626379523688210897945799582, 0.794588120318086899729737301335, 1.01672450645728970984382158112, 1.64982597012910951397424024223, 2.13188469695963530738113665950, 2.23701096439255923637237990751, 2.95874398382912614385956150552, 2.96949962833681951052352765362, 3.18750182489063981503494188642, 3.73475127956385771274174590578, 3.80888077538634852637200363057, 4.13712067953685228773405927587, 4.36173573418078513277175123221, 4.70527548286844812828528285987, 4.74945753477362831722670801586, 4.87168406933367139638899198053, 5.57586001860482077141586743162, 5.62290696045913103994483902539, 5.71047682058079076764186071235, 6.27201680063656869747919254524, 6.38673286049943538830405322446, 6.49504209158974204972098514929, 7.01504509627684399848489245223, 7.01788168277708826106803488149, 7.10446625739659078882182656826
