| L(s) = 1 | − 8·3-s − 32·9-s + 56·11-s + 128·17-s − 408·19-s + 12·25-s + 488·27-s − 448·33-s − 984·41-s − 856·43-s − 348·49-s − 1.02e3·51-s + 3.26e3·57-s − 2.45e3·59-s − 1.30e3·67-s − 1.66e3·73-s − 96·75-s − 466·81-s − 1.41e3·83-s − 128·89-s − 384·97-s − 1.79e3·99-s − 136·107-s + 1.08e3·113-s − 3.36e3·121-s + 7.87e3·123-s + 127-s + ⋯ |
| L(s) = 1 | − 1.53·3-s − 1.18·9-s + 1.53·11-s + 1.82·17-s − 4.92·19-s + 0.0959·25-s + 3.47·27-s − 2.36·33-s − 3.74·41-s − 3.03·43-s − 1.01·49-s − 2.81·51-s + 7.58·57-s − 5.41·59-s − 2.37·67-s − 2.66·73-s − 0.147·75-s − 0.639·81-s − 1.87·83-s − 0.152·89-s − 0.401·97-s − 1.81·99-s − 0.122·107-s + 0.899·113-s − 2.52·121-s + 5.77·123-s + 0.000698·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( ( 1 + 4 T + 40 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 5 | $C_2^2:C_4$ | \( 1 - 12 T^{2} - 1482 T^{4} - 12 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $C_2^2:C_4$ | \( 1 + 348 T^{2} + 134502 T^{4} + 348 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 28 T + 2856 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 84 T^{2} + 8049750 T^{4} + 84 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 64 T + 10050 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 204 T + 23784 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2:C_4$ | \( 1 + 31260 T^{2} + 471031590 T^{4} + 31260 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 60180 T^{2} + 1747419030 T^{4} + 60180 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 32388 T^{2} + 927859590 T^{4} - 32388 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 152948 T^{2} + 10364702902 T^{4} + 152948 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 12 p T + 165590 T^{2} + 12 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 428 T + 182760 T^{2} + 428 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 333372 T^{2} + 47698486086 T^{4} + 333372 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 4580 p T^{2} + 48450398710 T^{4} + 4580 p^{7} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 1228 T + 746856 T^{2} + 1228 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 133268 T^{2} + 27019375126 T^{4} + 133268 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 652 T + 488680 T^{2} + 652 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 300124 T^{2} + 278020688998 T^{4} + 300124 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 832 T + 948498 T^{2} + 832 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 886716 T^{2} + 560936541894 T^{4} + 886716 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 708 T + 1254440 T^{2} + 708 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 64 T + 1327730 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 192 T + 1318434 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−8.100376616903832128954621369925, −7.72130273684801326007179013366, −7.26960061116920398198756341356, −7.17288449997002545207229527749, −6.67713864943234285921499625106, −6.42749605106880587841216332064, −6.36701104634615323487314445468, −6.33865909158476914185239715613, −6.09698325949051673081770273222, −5.79401566820886115424165697237, −5.51491951188476719240306131067, −5.22636395990595923636775650974, −4.98589061422647040117625312338, −4.65889883576930890098345494346, −4.41733789288825021015428226512, −4.30208577556841827844855849165, −3.86657064795309983808159505819, −3.43151864822570691258748580147, −3.13857212134857562822294204991, −2.97232393209172952166468714818, −2.80628256693732701741612314752, −1.83129241970571230915066941041, −1.80924657747726942141363071253, −1.59574851753130292943671201302, −1.17001038829035988542607358971, 0, 0, 0, 0,
1.17001038829035988542607358971, 1.59574851753130292943671201302, 1.80924657747726942141363071253, 1.83129241970571230915066941041, 2.80628256693732701741612314752, 2.97232393209172952166468714818, 3.13857212134857562822294204991, 3.43151864822570691258748580147, 3.86657064795309983808159505819, 4.30208577556841827844855849165, 4.41733789288825021015428226512, 4.65889883576930890098345494346, 4.98589061422647040117625312338, 5.22636395990595923636775650974, 5.51491951188476719240306131067, 5.79401566820886115424165697237, 6.09698325949051673081770273222, 6.33865909158476914185239715613, 6.36701104634615323487314445468, 6.42749605106880587841216332064, 6.67713864943234285921499625106, 7.17288449997002545207229527749, 7.26960061116920398198756341356, 7.72130273684801326007179013366, 8.100376616903832128954621369925
