| L(s) = 1 | + 4-s − 12·11-s + 4·13-s − 8·17-s + 12·19-s + 4·23-s − 2·25-s + 4·29-s + 18·37-s − 10·43-s − 12·44-s − 10·49-s + 4·52-s + 24·53-s − 12·59-s − 64-s − 12·67-s − 8·68-s + 36·71-s + 12·76-s − 28·79-s − 12·89-s + 4·92-s − 12·97-s − 2·100-s − 8·101-s + 8·103-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 3.61·11-s + 1.10·13-s − 1.94·17-s + 2.75·19-s + 0.834·23-s − 2/5·25-s + 0.742·29-s + 2.95·37-s − 1.52·43-s − 1.80·44-s − 1.42·49-s + 0.554·52-s + 3.29·53-s − 1.56·59-s − 1/8·64-s − 1.46·67-s − 0.970·68-s + 4.27·71-s + 1.37·76-s − 3.15·79-s − 1.27·89-s + 0.417·92-s − 1.21·97-s − 1/5·100-s − 0.796·101-s + 0.788·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.927357081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.927357081\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| good | 7 | $C_2^3$ | \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 12 T + 73 T^{2} + 300 T^{3} + 1032 T^{4} + 300 p T^{5} + 73 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 12 T + 82 T^{2} - 408 T^{3} + 1707 T^{4} - 408 p T^{5} + 82 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 4 T - 31 T^{2} - 4 T^{3} + 1312 T^{4} - 4 p T^{5} - 31 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 4 T - 43 T^{2} - 4 T^{3} + 2176 T^{4} - 4 p T^{5} - 43 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 18 T + 193 T^{2} - 1530 T^{3} + 9852 T^{4} - 1530 p T^{5} + 193 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 78 T^{2} + 4403 T^{4} + 78 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 10 T + 37 T^{2} - 230 T^{3} - 2180 T^{4} - 230 p T^{5} + 37 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 66 T^{2} + 3155 T^{4} - 66 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 153 T^{2} + 1260 T^{3} + 10376 T^{4} + 1260 p T^{5} + 153 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^3$ | \( 1 - 14 T^{2} - 3525 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 12 T + 130 T^{2} + 984 T^{3} + 5451 T^{4} + 984 p T^{5} + 130 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 36 T + 678 T^{2} - 8856 T^{3} + 86147 T^{4} - 8856 p T^{5} + 678 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 14 T + 159 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 228 T^{2} + 26006 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T + 222 T^{2} + 2088 T^{3} + 26627 T^{4} + 2088 p T^{5} + 222 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 12 T + 238 T^{2} + 2280 T^{3} + 31347 T^{4} + 2280 p T^{5} + 238 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.12422473548658492877037738453, −6.69649808659619107169699267219, −6.59009554333405846030602547046, −6.43457967182876006210417657943, −5.98145080846933360432911491234, −5.83249634859988207577819728359, −5.63941808434867767800243329490, −5.38896975615607281424175572036, −5.34056642445280605196542758014, −4.90437790900253374430228014299, −4.73696865178550778279771079236, −4.67950811094152170966627093169, −4.26013888597885307488309349793, −3.99714080600462324151895101523, −3.72130937746582561100770843859, −3.15803039253591244165326822287, −3.09536516962179707959667223381, −2.84514183161256091429447935564, −2.78660213021323231316526761286, −2.34476944047085118245033778104, −2.15445974064712293644801157202, −1.69914196548426895204772195208, −1.31760104872918822249271101831, −0.67071398504075473525086385681, −0.49960030530360120169901087369,
0.49960030530360120169901087369, 0.67071398504075473525086385681, 1.31760104872918822249271101831, 1.69914196548426895204772195208, 2.15445974064712293644801157202, 2.34476944047085118245033778104, 2.78660213021323231316526761286, 2.84514183161256091429447935564, 3.09536516962179707959667223381, 3.15803039253591244165326822287, 3.72130937746582561100770843859, 3.99714080600462324151895101523, 4.26013888597885307488309349793, 4.67950811094152170966627093169, 4.73696865178550778279771079236, 4.90437790900253374430228014299, 5.34056642445280605196542758014, 5.38896975615607281424175572036, 5.63941808434867767800243329490, 5.83249634859988207577819728359, 5.98145080846933360432911491234, 6.43457967182876006210417657943, 6.59009554333405846030602547046, 6.69649808659619107169699267219, 7.12422473548658492877037738453
