| L(s) = 1 | − 4·3-s + 8·7-s + 8·9-s + 8·17-s − 32·21-s − 12·27-s + 8·43-s + 16·49-s − 32·51-s − 8·53-s − 16·59-s + 64·63-s + 24·67-s − 16·71-s + 23·81-s − 8·103-s − 48·109-s + 40·113-s + 64·119-s − 28·121-s + 127-s − 32·129-s + 131-s + 137-s + 139-s − 64·147-s + 149-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 3.02·7-s + 8/3·9-s + 1.94·17-s − 6.98·21-s − 2.30·27-s + 1.21·43-s + 16/7·49-s − 4.48·51-s − 1.09·53-s − 2.08·59-s + 8.06·63-s + 2.93·67-s − 1.89·71-s + 23/9·81-s − 0.788·103-s − 4.59·109-s + 3.76·113-s + 5.86·119-s − 2.54·121-s + 0.0887·127-s − 2.81·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.27·147-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8473810095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8473810095\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( ( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_4$ | \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 834 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 12 T^{2} - 1834 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 4 T - 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 144 T^{2} + 9314 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 12 T + 120 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_4\times C_2$ | \( 1 - 28 T^{2} - 1946 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 12774 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 224 T^{2} + 25522 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 8806 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_4\times C_2$ | \( 1 + 196 T^{2} + 23814 T^{4} + 196 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.29128611946578400354296671367, −6.15633685258650339456912781258, −5.75258851612972036423555921501, −5.56057890921579807123611140580, −5.54669801938641348090375748741, −5.29167511109410310883801434260, −5.22242817383124026907143615116, −5.11013596066189351990721417471, −4.72464907602739426994047339812, −4.51754276928168147820965184374, −4.31983889551441600072248925778, −4.24739278625272020089633189896, −4.10893663310484906126900559250, −3.61344274228210568767285987272, −3.17230813116482150705144574400, −3.11919692432197586783879417585, −3.04586055157771269956006818182, −2.36124828219222574285110202621, −2.07591076373122112024599757177, −1.83631285730619441152517095766, −1.68168275885937675116006372775, −1.21639590421627466315868786574, −1.08958214495032153731075557306, −0.899314813751426064777506090720, −0.18558646734476287946374585116,
0.18558646734476287946374585116, 0.899314813751426064777506090720, 1.08958214495032153731075557306, 1.21639590421627466315868786574, 1.68168275885937675116006372775, 1.83631285730619441152517095766, 2.07591076373122112024599757177, 2.36124828219222574285110202621, 3.04586055157771269956006818182, 3.11919692432197586783879417585, 3.17230813116482150705144574400, 3.61344274228210568767285987272, 4.10893663310484906126900559250, 4.24739278625272020089633189896, 4.31983889551441600072248925778, 4.51754276928168147820965184374, 4.72464907602739426994047339812, 5.11013596066189351990721417471, 5.22242817383124026907143615116, 5.29167511109410310883801434260, 5.54669801938641348090375748741, 5.56057890921579807123611140580, 5.75258851612972036423555921501, 6.15633685258650339456912781258, 6.29128611946578400354296671367
