| L(s) = 1 | − 4·2-s + 3·3-s + 10·4-s − 12·6-s − 28·7-s − 17·8-s − 19·9-s + 100·11-s + 30·12-s − 44·13-s + 112·14-s − 10·16-s + 53·17-s + 76·18-s − 29·19-s − 84·21-s − 400·22-s − 295·23-s − 51·24-s + 176·26-s + 54·27-s − 280·28-s + 129·29-s + 114·31-s + 212·32-s + 300·33-s − 212·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 5/4·4-s − 0.816·6-s − 1.51·7-s − 0.751·8-s − 0.703·9-s + 2.74·11-s + 0.721·12-s − 0.938·13-s + 2.13·14-s − 0.156·16-s + 0.756·17-s + 0.995·18-s − 0.350·19-s − 0.872·21-s − 3.87·22-s − 2.67·23-s − 0.433·24-s + 1.32·26-s + 0.384·27-s − 1.88·28-s + 0.826·29-s + 0.660·31-s + 1.17·32-s + 1.58·33-s − 1.06·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.465152078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.465152078\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 + p^{2} T + 3 p T^{2} + T^{3} + 11 p T^{4} + p^{3} T^{5} + 3 p^{7} T^{6} + p^{11} T^{7} + p^{12} T^{8} \) |
| 3 | $C_2 \wr S_4$ | \( 1 - p T + 28 T^{2} - 65 p T^{3} + 1150 T^{4} - 65 p^{4} T^{5} + 28 p^{6} T^{6} - p^{10} T^{7} + p^{12} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 100 T + 4738 T^{2} - 106848 T^{3} + 2242403 T^{4} - 106848 p^{3} T^{5} + 4738 p^{6} T^{6} - 100 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 44 T + 4820 T^{2} + 154316 T^{3} + 10485014 T^{4} + 154316 p^{3} T^{5} + 4820 p^{6} T^{6} + 44 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 53 T + 15496 T^{2} - 576511 T^{3} + 103902926 T^{4} - 576511 p^{3} T^{5} + 15496 p^{6} T^{6} - 53 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 29 T + 6638 T^{2} + 258553 T^{3} + 53805642 T^{4} + 258553 p^{3} T^{5} + 6638 p^{6} T^{6} + 29 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 295 T + 54367 T^{2} + 7704248 T^{3} + 887991584 T^{4} + 7704248 p^{3} T^{5} + 54367 p^{6} T^{6} + 295 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 129 T + 83933 T^{2} - 9167698 T^{3} + 2922882282 T^{4} - 9167698 p^{3} T^{5} + 83933 p^{6} T^{6} - 129 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 114 T + 80728 T^{2} - 5927082 T^{3} + 3079701134 T^{4} - 5927082 p^{3} T^{5} + 80728 p^{6} T^{6} - 114 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 403 T + 202319 T^{2} + 46625606 T^{3} + 14127493020 T^{4} + 46625606 p^{3} T^{5} + 202319 p^{6} T^{6} + 403 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 671 T + 399120 T^{2} - 139154877 T^{3} + 44432315734 T^{4} - 139154877 p^{3} T^{5} + 399120 p^{6} T^{6} - 671 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 411 T + 213439 T^{2} - 60368868 T^{3} + 23440158536 T^{4} - 60368868 p^{3} T^{5} + 213439 p^{6} T^{6} - 411 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 8 T + 160820 T^{2} + 20939512 T^{3} + 12301264614 T^{4} + 20939512 p^{3} T^{5} + 160820 p^{6} T^{6} - 8 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 90 T + 133752 T^{2} - 19514962 T^{3} + 19633648894 T^{4} - 19514962 p^{3} T^{5} + 133752 p^{6} T^{6} + 90 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 1018 T + 1189512 T^{2} - 683319466 T^{3} + 407279566174 T^{4} - 683319466 p^{3} T^{5} + 1189512 p^{6} T^{6} - 1018 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 50 T + 722188 T^{2} - 51306118 T^{3} + 225994176278 T^{4} - 51306118 p^{3} T^{5} + 722188 p^{6} T^{6} - 50 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 424 T + 917806 T^{2} + 255868016 T^{3} + 369880128047 T^{4} + 255868016 p^{3} T^{5} + 917806 p^{6} T^{6} + 424 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 215 T + 982021 T^{2} - 275350320 T^{3} + 449702043326 T^{4} - 275350320 p^{3} T^{5} + 982021 p^{6} T^{6} - 215 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 1207 T + 1581740 T^{2} - 1193773973 T^{3} + 948033109470 T^{4} - 1193773973 p^{3} T^{5} + 1581740 p^{6} T^{6} - 1207 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 951 T + 1431057 T^{2} + 665911232 T^{3} + 746318767554 T^{4} + 665911232 p^{3} T^{5} + 1431057 p^{6} T^{6} + 951 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 3035 T + 5488190 T^{2} - 6563721455 T^{3} + 5816729995458 T^{4} - 6563721455 p^{3} T^{5} + 5488190 p^{6} T^{6} - 3035 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 2819 T + 5531118 T^{2} - 7041312813 T^{3} + 6955665197242 T^{4} - 7041312813 p^{3} T^{5} + 5531118 p^{6} T^{6} - 2819 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 1100 T + 625060 T^{2} + 1205286580 T^{3} - 1327506690522 T^{4} + 1205286580 p^{3} T^{5} + 625060 p^{6} T^{6} - 1100 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−9.096500229355524334596048729139, −8.556627141196053326647380862039, −8.289592867360317828739872464393, −8.063514353850715638176607023040, −8.043101483122425136014638530171, −7.37911752003919553288537713499, −7.31327697550779981426658038184, −6.75822527144168371605956012013, −6.73154344767138228439247298320, −6.45435649022378330035426048120, −6.07923502038472262505001794579, −5.89156880718892164262276887496, −5.73767964529781460104708135065, −4.93650334657927347687269860537, −4.67579796851340841731909011122, −4.13589209371344773455849592961, −3.94033848476198873690756859061, −3.50956635581667309636234536906, −3.38116872597316928849224752835, −2.57098019204571101580261907459, −2.42601663334762778329942701282, −2.11137830062807610517246830535, −1.31829429226928772840387860829, −0.810811962771265949365583456405, −0.41960715885990573395412704509,
0.41960715885990573395412704509, 0.810811962771265949365583456405, 1.31829429226928772840387860829, 2.11137830062807610517246830535, 2.42601663334762778329942701282, 2.57098019204571101580261907459, 3.38116872597316928849224752835, 3.50956635581667309636234536906, 3.94033848476198873690756859061, 4.13589209371344773455849592961, 4.67579796851340841731909011122, 4.93650334657927347687269860537, 5.73767964529781460104708135065, 5.89156880718892164262276887496, 6.07923502038472262505001794579, 6.45435649022378330035426048120, 6.73154344767138228439247298320, 6.75822527144168371605956012013, 7.31327697550779981426658038184, 7.37911752003919553288537713499, 8.043101483122425136014638530171, 8.063514353850715638176607023040, 8.289592867360317828739872464393, 8.556627141196053326647380862039, 9.096500229355524334596048729139
