| L(s) = 1 | − 2·3-s + 2·5-s + 5·9-s + 4·11-s − 8·13-s − 4·15-s − 4·17-s + 14·23-s + 25-s − 10·27-s − 20·29-s − 4·31-s − 8·33-s + 16·39-s − 12·41-s + 12·43-s + 10·45-s + 12·47-s + 8·51-s + 8·55-s − 8·59-s + 2·61-s − 16·65-s + 6·67-s − 28·69-s − 48·71-s − 4·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 5/3·9-s + 1.20·11-s − 2.21·13-s − 1.03·15-s − 0.970·17-s + 2.91·23-s + 1/5·25-s − 1.92·27-s − 3.71·29-s − 0.718·31-s − 1.39·33-s + 2.56·39-s − 1.87·41-s + 1.82·43-s + 1.49·45-s + 1.75·47-s + 1.12·51-s + 1.07·55-s − 1.04·59-s + 0.256·61-s − 1.98·65-s + 0.733·67-s − 3.37·69-s − 5.69·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.877736991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.877736991\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 T - T^{2} - 2 T^{3} + 4 T^{4} - 2 p T^{5} - p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4 T - 2 T^{2} + 16 T^{3} + 27 T^{4} + 16 p T^{5} - 2 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 17 | $C_4\times C_2$ | \( 1 + 4 T + 10 T^{2} - 112 T^{3} - 525 T^{4} - 112 p T^{5} + 10 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 - 6 T^{2} - 325 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 14 T + 103 T^{2} - 658 T^{3} + 3612 T^{4} - 658 p T^{5} + 103 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 10 T + 75 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 4 T - 42 T^{2} - 16 T^{3} + 1907 T^{4} - 16 p T^{5} - 42 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 - 42 T^{2} + 395 T^{4} - 42 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 6 T - 3 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 12 T + 46 T^{2} - 48 T^{3} + 627 T^{4} - 48 p T^{5} + 46 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 74 T^{2} + 2667 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 2 T - 87 T^{2} + 62 T^{3} + 4316 T^{4} + 62 p T^{5} - 87 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6 T - 9 T^{2} + 534 T^{3} - 4876 T^{4} + 534 p T^{5} - 9 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 + 4 T - 102 T^{2} - 112 T^{3} + 7427 T^{4} - 112 p T^{5} - 102 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 18 T + 229 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 22 T + 217 T^{2} + 22 p T^{3} + 228 p T^{4} + 22 p^{2} T^{5} + 217 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
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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
−6.60358104084779261442531334988, −6.43248918749161162828341245657, −5.86871739422377481973774961366, −5.82144114676232867457707226665, −5.64655965797221910996436744012, −5.60101351024758720380747272725, −5.42256140193026794703822297835, −4.94961150681286790626784460892, −4.80683897202288295978781444565, −4.63130382762962622319403544108, −4.60161804479991093767367404680, −4.15025242153619191721745797788, −3.99850507132720828686530795474, −3.66005570370460033093663051106, −3.62953137543969708399175099860, −3.02685226922706435411782475494, −2.95792377415495318627261965315, −2.55735138254886223937107574474, −2.39045374121638507053395795534, −1.82965507482235433738511637111, −1.73179142722232328469696462737, −1.65333404675059905987554940209, −1.22475454935689919639932097588, −0.58286117478084295658072423903, −0.34941546957558404290195694030,
0.34941546957558404290195694030, 0.58286117478084295658072423903, 1.22475454935689919639932097588, 1.65333404675059905987554940209, 1.73179142722232328469696462737, 1.82965507482235433738511637111, 2.39045374121638507053395795534, 2.55735138254886223937107574474, 2.95792377415495318627261965315, 3.02685226922706435411782475494, 3.62953137543969708399175099860, 3.66005570370460033093663051106, 3.99850507132720828686530795474, 4.15025242153619191721745797788, 4.60161804479991093767367404680, 4.63130382762962622319403544108, 4.80683897202288295978781444565, 4.94961150681286790626784460892, 5.42256140193026794703822297835, 5.60101351024758720380747272725, 5.64655965797221910996436744012, 5.82144114676232867457707226665, 5.86871739422377481973774961366, 6.43248918749161162828341245657, 6.60358104084779261442531334988
