| L(s) = 1 | − 3-s + 4·5-s + 8·7-s + 3·9-s − 4·15-s + 6·17-s − 8·21-s + 10·25-s − 8·27-s + 32·35-s − 4·37-s − 24·41-s + 4·43-s + 12·45-s + 18·47-s + 34·49-s − 6·51-s + 12·59-s + 24·63-s + 4·67-s − 10·75-s − 2·79-s + 8·81-s + 24·83-s + 24·85-s + 36·89-s + 24·101-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s + 3.02·7-s + 9-s − 1.03·15-s + 1.45·17-s − 1.74·21-s + 2·25-s − 1.53·27-s + 5.40·35-s − 0.657·37-s − 3.74·41-s + 0.609·43-s + 1.78·45-s + 2.62·47-s + 34/7·49-s − 0.840·51-s + 1.56·59-s + 3.02·63-s + 0.488·67-s − 1.15·75-s − 0.225·79-s + 8/9·81-s + 2.63·83-s + 2.60·85-s + 3.81·89-s + 2.38·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.61160921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.61160921\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| good | 11 | $D_4\times C_2$ | \( 1 - 37 T^{2} + 576 T^{4} - 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - T^{2} + 264 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 16 T^{2} - 66 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 97 T^{2} + 3960 T^{4} - 97 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 43 | $D_{4}$ | \( ( 1 - 2 T + 54 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 9 T + 106 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 136 T^{2} + 9054 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4134 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + T + 150 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 12 T + 70 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 18 T + 226 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 265 T^{2} + 32736 T^{4} - 265 p^{2} T^{6} + p^{4} T^{8} \) |
| 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
−6.70384199849461612481332392893, −6.38009393125032739416427796896, −6.19580106160943488672759150192, −5.84722953507496251675542112664, −5.81164826170481572392949368662, −5.43541688624429952774164746025, −5.31345107649271014879651025619, −5.25909993946880658407176896038, −4.97434084130805895185228047897, −4.89054616005883675093584284413, −4.60169155985136736104765111665, −4.35352056821340304696296510715, −4.06738347388269426945894757292, −3.74834876046584214026066090217, −3.63943847877238736800803743141, −3.19302991078330830961033496169, −3.08276312176775623982388886195, −2.48347355948800679712497443321, −2.16337322651037326053200198931, −1.97300157660937276103089559676, −1.82793391029484546896910992866, −1.76258207761313104480763683610, −1.17380517002660459494909493075, −0.949243845249086983080356334854, −0.69300647530436120233477833381,
0.69300647530436120233477833381, 0.949243845249086983080356334854, 1.17380517002660459494909493075, 1.76258207761313104480763683610, 1.82793391029484546896910992866, 1.97300157660937276103089559676, 2.16337322651037326053200198931, 2.48347355948800679712497443321, 3.08276312176775623982388886195, 3.19302991078330830961033496169, 3.63943847877238736800803743141, 3.74834876046584214026066090217, 4.06738347388269426945894757292, 4.35352056821340304696296510715, 4.60169155985136736104765111665, 4.89054616005883675093584284413, 4.97434084130805895185228047897, 5.25909993946880658407176896038, 5.31345107649271014879651025619, 5.43541688624429952774164746025, 5.81164826170481572392949368662, 5.84722953507496251675542112664, 6.19580106160943488672759150192, 6.38009393125032739416427796896, 6.70384199849461612481332392893
