| L(s) = 1 | + 8·9-s − 10·25-s − 24·29-s + 48·41-s − 8·49-s + 30·81-s + 72·101-s − 44·121-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 | + 8/3·9-s − 2·25-s − 4.45·29-s + 7.49·41-s − 8/7·49-s + 10/3·81-s + 7.16·101-s − 4·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.169088500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.169088500\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
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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
−6.53030672977214159267916621211, −6.33830228616134470610451434397, −6.05332309495927469186098916214, −5.95054161269808339431890328114, −5.87828133552865796052554823830, −5.49144701961845764894916238254, −5.44125879953846344446454867939, −4.98576917281967103727691240206, −4.94654517481940278330848550806, −4.32511251589903727092955923506, −4.24622284453921309041975398225, −4.22353258160984330676185182317, −4.07293377587954730748701419088, −3.97095303718592121737034638264, −3.36165500083336533606413305160, −3.29590481158386921081225895181, −3.20672562077088951890911286649, −2.35789569964417640388409082685, −2.26038205265691797041900241180, −2.12864841305706741144718164091, −1.98267923644825071147193256303, −1.50523716604757432411317030971, −1.05077269191663283650414639431, −0.992719158316320777705041565287, −0.31706902330945866442491259851,
0.31706902330945866442491259851, 0.992719158316320777705041565287, 1.05077269191663283650414639431, 1.50523716604757432411317030971, 1.98267923644825071147193256303, 2.12864841305706741144718164091, 2.26038205265691797041900241180, 2.35789569964417640388409082685, 3.20672562077088951890911286649, 3.29590481158386921081225895181, 3.36165500083336533606413305160, 3.97095303718592121737034638264, 4.07293377587954730748701419088, 4.22353258160984330676185182317, 4.24622284453921309041975398225, 4.32511251589903727092955923506, 4.94654517481940278330848550806, 4.98576917281967103727691240206, 5.44125879953846344446454867939, 5.49144701961845764894916238254, 5.87828133552865796052554823830, 5.95054161269808339431890328114, 6.05332309495927469186098916214, 6.33830228616134470610451434397, 6.53030672977214159267916621211
