| L(s) = 1 | + 2·3-s + 4-s + 4·9-s + 2·12-s − 6·17-s + 12·19-s − 12·23-s − 2·25-s + 4·27-s + 12·29-s + 4·36-s − 18·37-s + 36·41-s − 4·43-s − 5·49-s − 12·51-s − 12·53-s + 24·57-s − 36·59-s − 2·61-s − 64-s − 6·68-s − 24·69-s − 4·75-s + 12·76-s + 4·79-s + 5·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 4/3·9-s + 0.577·12-s − 1.45·17-s + 2.75·19-s − 2.50·23-s − 2/5·25-s + 0.769·27-s + 2.22·29-s + 2/3·36-s − 2.95·37-s + 5.62·41-s − 0.609·43-s − 5/7·49-s − 1.68·51-s − 1.64·53-s + 3.17·57-s − 4.68·59-s − 0.256·61-s − 1/8·64-s − 0.727·68-s − 2.88·69-s − 0.461·75-s + 1.37·76-s + 0.450·79-s + 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.482835989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.482835989\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 T + 4 T^{3} - 5 T^{4} + 4 p T^{5} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^3$ | \( 1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 6 T + 20 T^{2} - 108 T^{3} - 645 T^{4} - 108 p T^{5} + 20 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 12 T + 89 T^{2} - 492 T^{3} + 2232 T^{4} - 492 p T^{5} + 89 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 12 T + 74 T^{2} + 288 T^{3} + 1059 T^{4} + 288 p T^{5} + 74 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 12 T + 62 T^{2} - 288 T^{3} + 1707 T^{4} - 288 p T^{5} + 62 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4314 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 18 T + 173 T^{2} + 1170 T^{3} + 6852 T^{4} + 1170 p T^{5} + 173 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 18 T + 149 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 6 T + 103 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 18 T + 167 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 2 T - 92 T^{2} - 52 T^{3} + 5251 T^{4} - 52 p T^{5} - 92 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^3$ | \( 1 + 106 T^{2} + 6195 T^{4} + 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 11802 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 2 T + 132 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 29706 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 18 T + 169 T^{2} - 1098 T^{3} + 5412 T^{4} - 1098 p T^{5} + 169 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 42 T + 920 T^{2} + 13944 T^{3} + 157851 T^{4} + 13944 p T^{5} + 920 p^{2} T^{6} + 42 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.49568993019805168759082254544, −6.47282071809513532850166004374, −6.43361644048266595437449235010, −6.11678839380510351796648637823, −5.74514745990836504979174473998, −5.68812772276304152422557023025, −5.40211725810051437105435401527, −5.03666706566587814907778903519, −4.80015918454119117945836965079, −4.78763656256207063173675766592, −4.38353007987954978905711890612, −4.08421821567275985356975623442, −3.98608395646959901098930521370, −3.76209546831043714624299528741, −3.62759743551168212523366692970, −3.01090587160604180286135945189, −2.89124411588070991735693284193, −2.75339008262501818742218440716, −2.58569071672347110502669688552, −2.32162378615252066444111136693, −1.57987442111881741198968819990, −1.56516053968533181871759089463, −1.55279970216025816246411512947, −0.998263171850622820431014178808, −0.17569834611816404218082576969,
0.17569834611816404218082576969, 0.998263171850622820431014178808, 1.55279970216025816246411512947, 1.56516053968533181871759089463, 1.57987442111881741198968819990, 2.32162378615252066444111136693, 2.58569071672347110502669688552, 2.75339008262501818742218440716, 2.89124411588070991735693284193, 3.01090587160604180286135945189, 3.62759743551168212523366692970, 3.76209546831043714624299528741, 3.98608395646959901098930521370, 4.08421821567275985356975623442, 4.38353007987954978905711890612, 4.78763656256207063173675766592, 4.80015918454119117945836965079, 5.03666706566587814907778903519, 5.40211725810051437105435401527, 5.68812772276304152422557023025, 5.74514745990836504979174473998, 6.11678839380510351796648637823, 6.43361644048266595437449235010, 6.47282071809513532850166004374, 6.49568993019805168759082254544
