L(s) = 1 | + 2·3-s − 4·5-s + 2·9-s − 8·15-s − 20·19-s + 16·23-s + 6·27-s − 16·43-s − 8·45-s − 8·47-s − 2·49-s + 24·53-s − 40·57-s + 8·67-s + 32·69-s + 8·71-s + 16·73-s + 11·81-s + 80·95-s + 32·97-s + 52·101-s − 64·115-s + 4·121-s + 20·125-s + 127-s − 32·129-s + 131-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 1.78·5-s + 2/3·9-s − 2.06·15-s − 4.58·19-s + 3.33·23-s + 1.15·27-s − 2.43·43-s − 1.19·45-s − 1.16·47-s − 2/7·49-s + 3.29·53-s − 5.29·57-s + 0.977·67-s + 3.85·69-s + 0.949·71-s + 1.87·73-s + 11/9·81-s + 8.20·95-s + 3.24·97-s + 5.17·101-s − 5.96·115-s + 4/11·121-s + 1.78·125-s + 0.0887·127-s − 2.81·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.304364281\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.304364281\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
good | 5 | $D_{4}$ | \( ( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 718 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4918 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 224 T^{2} + 19486 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 104 T^{2} + 5646 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_4$ | \( ( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 11206 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 12958 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.62053682340410561458658912939, −7.40593661574741480281690622070, −6.99616398017840697550945654650, −6.95150025216281721892354961290, −6.70089671891400740485751396464, −6.52124110509473423729468183206, −6.39058306080817616759715728943, −5.87121081466773173217164481318, −5.80602703005546441285677486945, −5.11269754387412842185460199688, −5.05797169773987507887363006590, −4.81999095681755374210614463868, −4.52410461185967835691823544576, −4.30642549090411250958312298262, −4.06946974847285981270663644857, −3.81251059965010257535930746383, −3.32665850844414997549934612555, −3.28417849310778778065760574609, −3.25622851948429840842139211934, −2.48899716983736751995079243038, −2.23694824916424707895718832613, −2.02790887617416173862735005941, −1.77102092293634023788360117392, −0.68265374875713605552763993352, −0.58809778651350895198125767938,
0.58809778651350895198125767938, 0.68265374875713605552763993352, 1.77102092293634023788360117392, 2.02790887617416173862735005941, 2.23694824916424707895718832613, 2.48899716983736751995079243038, 3.25622851948429840842139211934, 3.28417849310778778065760574609, 3.32665850844414997549934612555, 3.81251059965010257535930746383, 4.06946974847285981270663644857, 4.30642549090411250958312298262, 4.52410461185967835691823544576, 4.81999095681755374210614463868, 5.05797169773987507887363006590, 5.11269754387412842185460199688, 5.80602703005546441285677486945, 5.87121081466773173217164481318, 6.39058306080817616759715728943, 6.52124110509473423729468183206, 6.70089671891400740485751396464, 6.95150025216281721892354961290, 6.99616398017840697550945654650, 7.40593661574741480281690622070, 7.62053682340410561458658912939