| L(s) = 1 | + 2·3-s + 9-s + 6·11-s + 4·13-s − 2·17-s + 4·19-s + 4·23-s − 2·27-s + 8·29-s − 8·31-s + 12·33-s − 6·37-s + 8·39-s − 12·41-s − 8·43-s + 10·47-s − 7·49-s − 4·51-s − 10·53-s + 8·57-s − 14·59-s + 2·61-s + 8·69-s − 8·71-s + 2·73-s + 8·79-s − 4·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 1.80·11-s + 1.10·13-s − 0.485·17-s + 0.917·19-s + 0.834·23-s − 0.384·27-s + 1.48·29-s − 1.43·31-s + 2.08·33-s − 0.986·37-s + 1.28·39-s − 1.87·41-s − 1.21·43-s + 1.45·47-s − 49-s − 0.560·51-s − 1.37·53-s + 1.05·57-s − 1.82·59-s + 0.256·61-s + 0.963·69-s − 0.949·71-s + 0.234·73-s + 0.900·79-s − 4/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.966389214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.966389214\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| good | 11 | $D_4\times C_2$ | \( 1 - 6 T + 12 T^{2} - 12 T^{3} + 59 T^{4} - 12 p T^{5} + 12 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 17 | $D_4\times C_2$ | \( 1 + 2 T - 24 T^{2} - 12 T^{3} + 427 T^{4} - 12 p T^{5} - 24 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 4 T - p T^{2} + 12 T^{3} + 560 T^{4} + 12 p T^{5} - p^{3} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 6 T - 19 T^{2} - 114 T^{3} + 324 T^{4} - 114 p T^{5} - 19 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 6 T + 84 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 4 T + 62 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 10 T + 44 T^{2} + 380 T^{3} - 3773 T^{4} + 380 p T^{5} + 44 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 10 T - 24 T^{2} + 180 T^{3} + 7267 T^{4} + 180 p T^{5} - 24 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 14 T + 92 T^{2} - 196 T^{3} - 4229 T^{4} - 196 p T^{5} + 92 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 2 T - 7 T^{2} + 222 T^{3} - 3844 T^{4} + 222 p T^{5} - 7 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^3$ | \( 1 - 71 T^{2} + 552 T^{4} - 71 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 2 T - 115 T^{2} + 54 T^{3} + 8540 T^{4} + 54 p T^{5} - 115 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 8 T - 103 T^{2} - 72 T^{3} + 16592 T^{4} - 72 p T^{5} - 103 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 2 T + 160 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 2 T - 112 T^{2} + 124 T^{3} + 5179 T^{4} + 124 p T^{5} - 112 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 18 T + 247 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.61409711183414296675320110360, −6.22335607433833516580176663366, −6.11934907689168355795222916923, −5.86124544058263150574532025420, −5.70004076211337768864493972897, −5.56618599101925690771393402990, −4.97265109191210842081068454561, −4.88993037165904390080933231280, −4.74004054258605154199396875617, −4.67836377980628206571470607949, −4.34636790798637541610132161878, −3.93836873614270823478416721924, −3.69478294487846053654327893406, −3.51880662395650570328080120776, −3.32703150271986895432829098900, −3.23289565860442258193798858965, −3.20756718949532002485898824222, −2.48772151417875532885191817333, −2.48438950272887571113377428261, −2.02760359980690275902683012010, −1.65339262427500792761075656774, −1.50001483961399123400964042113, −1.39646693578156109908360905932, −0.810022429285912094020162092029, −0.40905904102802907762329329155,
0.40905904102802907762329329155, 0.810022429285912094020162092029, 1.39646693578156109908360905932, 1.50001483961399123400964042113, 1.65339262427500792761075656774, 2.02760359980690275902683012010, 2.48438950272887571113377428261, 2.48772151417875532885191817333, 3.20756718949532002485898824222, 3.23289565860442258193798858965, 3.32703150271986895432829098900, 3.51880662395650570328080120776, 3.69478294487846053654327893406, 3.93836873614270823478416721924, 4.34636790798637541610132161878, 4.67836377980628206571470607949, 4.74004054258605154199396875617, 4.88993037165904390080933231280, 4.97265109191210842081068454561, 5.56618599101925690771393402990, 5.70004076211337768864493972897, 5.86124544058263150574532025420, 6.11934907689168355795222916923, 6.22335607433833516580176663366, 6.61409711183414296675320110360
