| L(s) = 1 | − 4·7-s + 6·9-s − 16·17-s − 2·25-s − 16·31-s − 16·41-s − 12·47-s + 10·49-s − 24·63-s − 16·71-s + 16·73-s − 16·79-s + 9·81-s + 24·89-s − 64·97-s − 4·103-s − 8·113-s + 64·119-s + 42·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 96·153-s + 157-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 2·9-s − 3.88·17-s − 2/5·25-s − 2.87·31-s − 2.49·41-s − 1.75·47-s + 10/7·49-s − 3.02·63-s − 1.89·71-s + 1.87·73-s − 1.80·79-s + 81-s + 2.54·89-s − 6.49·97-s − 0.394·103-s − 0.752·113-s + 5.86·119-s + 3.81·121-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 − 7.76·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7156202737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7156202737\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 21 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 30 T^{2} + 179 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 4086 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 6 T + 55 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 180 T^{2} + 13526 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 10134 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 212 T^{2} + 18486 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 22710 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 8 T + 146 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 79 | $D_{4}$ | \( ( 1 + 8 T + 171 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 6966 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 12 T + 202 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 32 T + 447 T^{2} + 32 p T^{3} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.43709063223211639828688211839, −6.39453677587344729099058034542, −6.33542035466410140558724979677, −5.60503788220601188496672439681, −5.54597942805877710702293714472, −5.44457749653343814254518111843, −5.33092702040372285351025578045, −4.75301810028426929995802062698, −4.61155509960550874115648041315, −4.46026501271324119553621752662, −4.33490315593431359512513950388, −4.09401690912087438375180018375, −3.88408395173203319196690010167, −3.63249262191462742707065738404, −3.28573414485977212504911094074, −3.23023055982139221695062235033, −2.95797526962757474024310996566, −2.47246134159906935530965254245, −2.25948999928854333983943815111, −2.02905711773789271051580642974, −1.76363464228013742417460930245, −1.50130127120483805869682704475, −1.35279684342035068734097316451, −0.34036710664246468053305279547, −0.28562460193322746024976883790,
0.28562460193322746024976883790, 0.34036710664246468053305279547, 1.35279684342035068734097316451, 1.50130127120483805869682704475, 1.76363464228013742417460930245, 2.02905711773789271051580642974, 2.25948999928854333983943815111, 2.47246134159906935530965254245, 2.95797526962757474024310996566, 3.23023055982139221695062235033, 3.28573414485977212504911094074, 3.63249262191462742707065738404, 3.88408395173203319196690010167, 4.09401690912087438375180018375, 4.33490315593431359512513950388, 4.46026501271324119553621752662, 4.61155509960550874115648041315, 4.75301810028426929995802062698, 5.33092702040372285351025578045, 5.44457749653343814254518111843, 5.54597942805877710702293714472, 5.60503788220601188496672439681, 6.33542035466410140558724979677, 6.39453677587344729099058034542, 6.43709063223211639828688211839
