| L(s) = 1 | + 16·25-s + 8·37-s − 2·49-s + 56·61-s − 16·73-s + 48·97-s + 16·109-s − 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 16/5·25-s + 1.31·37-s − 2/7·49-s + 7.17·61-s − 1.87·73-s + 4.87·97-s + 1.53·109-s − 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.850916891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.850916891\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 64 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
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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
−6.72755229860105836148682577352, −6.13378797505346986127825967807, −6.10022941954816105002613787452, −5.97959861926028179555670356900, −5.77870953670976972409552112606, −5.22195282903061445921147115363, −5.11428380093113554571480692738, −5.09823086585336003569263552430, −4.98327593540632119025748412068, −4.61393723067559192617765375739, −4.32249950937067055866549690682, −4.08842498581511512078341643103, −3.97035255915457473901158743495, −3.53652878900449778367801757992, −3.48722627995772589993524231658, −3.09190710178931241593516484010, −2.99933668060109306797856618402, −2.49581092214053394736839167546, −2.43915829051214301324097507163, −2.14114183450065497506106846657, −1.96655879057002477742925990838, −1.26452439413478438209653479126, −0.994086586373839914937859809760, −0.919754085501487020320916097956, −0.43444252827474705432675170804,
0.43444252827474705432675170804, 0.919754085501487020320916097956, 0.994086586373839914937859809760, 1.26452439413478438209653479126, 1.96655879057002477742925990838, 2.14114183450065497506106846657, 2.43915829051214301324097507163, 2.49581092214053394736839167546, 2.99933668060109306797856618402, 3.09190710178931241593516484010, 3.48722627995772589993524231658, 3.53652878900449778367801757992, 3.97035255915457473901158743495, 4.08842498581511512078341643103, 4.32249950937067055866549690682, 4.61393723067559192617765375739, 4.98327593540632119025748412068, 5.09823086585336003569263552430, 5.11428380093113554571480692738, 5.22195282903061445921147115363, 5.77870953670976972409552112606, 5.97959861926028179555670356900, 6.10022941954816105002613787452, 6.13378797505346986127825967807, 6.72755229860105836148682577352
