L(s) = 1 | + 4·4-s + 4·5-s + 4·7-s + 7·16-s + 16·20-s + 10·25-s + 16·28-s + 16·35-s + 8·37-s + 8·43-s − 24·47-s − 2·49-s − 24·59-s + 8·64-s − 16·67-s − 16·79-s + 28·80-s + 24·83-s − 24·89-s + 40·100-s + 32·109-s + 28·112-s + 16·121-s + 20·125-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | + 2·4-s + 1.78·5-s + 1.51·7-s + 7/4·16-s + 3.57·20-s + 2·25-s + 3.02·28-s + 2.70·35-s + 1.31·37-s + 1.21·43-s − 3.50·47-s − 2/7·49-s − 3.12·59-s + 64-s − 1.95·67-s − 1.80·79-s + 3.13·80-s + 2.63·83-s − 2.54·89-s + 4·100-s + 3.06·109-s + 2.64·112-s + 1.45·121-s + 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.201869001\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.201869001\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
good | 2 | $D_4\times C_2$ | \( 1 - p^{2} T^{2} + 9 T^{4} - p^{4} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 114 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 90 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 486 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 2934 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 5826 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 12 T + 142 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1206 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 32 T^{2} + 78 p T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 244 T^{2} + 25110 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 12 T + 190 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 15606 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
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
−8.675618306953813845613530024764, −7.906440888919063439269924225338, −7.80977471373673174662320924812, −7.79442081431534446587376304985, −7.69874870166444998219624472639, −7.02810906694292914505111306980, −6.88171267493195176981269538222, −6.62681788617958233155747121512, −6.50067855380537693524795530447, −5.99628308600570897415193958069, −5.93738981558368653400554332536, −5.84476632100703426418965401639, −5.39477166136413814713265343408, −4.97165306204623213123493477573, −4.76529562969078925914512920827, −4.48060314309832582078767236639, −4.36218637575472330331694073788, −3.52459565625723560950418041350, −3.20171324690582010984728758088, −2.78724457082843949553297470231, −2.77003279710001407088362527445, −2.02453488071682227673094325807, −1.85199912483380229660003245258, −1.65645164193364391998557010203, −1.20480641101168406966098597349,
1.20480641101168406966098597349, 1.65645164193364391998557010203, 1.85199912483380229660003245258, 2.02453488071682227673094325807, 2.77003279710001407088362527445, 2.78724457082843949553297470231, 3.20171324690582010984728758088, 3.52459565625723560950418041350, 4.36218637575472330331694073788, 4.48060314309832582078767236639, 4.76529562969078925914512920827, 4.97165306204623213123493477573, 5.39477166136413814713265343408, 5.84476632100703426418965401639, 5.93738981558368653400554332536, 5.99628308600570897415193958069, 6.50067855380537693524795530447, 6.62681788617958233155747121512, 6.88171267493195176981269538222, 7.02810906694292914505111306980, 7.69874870166444998219624472639, 7.79442081431534446587376304985, 7.80977471373673174662320924812, 7.906440888919063439269924225338, 8.675618306953813845613530024764