L(s) = 1 | − 2·9-s − 4·17-s + 20·23-s + 12·25-s + 8·31-s + 4·41-s − 36·71-s − 48·73-s + 8·79-s + 3·81-s − 20·89-s + 32·97-s + 24·103-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 8·153-s + 157-s + 163-s + 167-s + 20·169-s + 173-s + 179-s + ⋯ |
L(s) = 1 | − 2/3·9-s − 0.970·17-s + 4.17·23-s + 12/5·25-s + 1.43·31-s + 0.624·41-s − 4.27·71-s − 5.61·73-s + 0.900·79-s + 1/3·81-s − 2.11·89-s + 3.24·97-s + 2.36·103-s + 1.81·121-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 + 0.646·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.53·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.278835506\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.278835506\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
good | 5 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 74 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 234 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 246 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 54 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 10 T + 68 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 2454 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6294 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 12 T^{2} - 5290 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 18 T + 220 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 24 T + 278 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 300 T^{2} + 36086 T^{4} - 300 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 10 T + 200 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 246 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−5.95052219106961059175235023150, −5.60189006570404819953216229041, −5.53812911921192988482297313241, −5.23373329469267946722818597361, −4.91138809342765049035436810025, −4.87397685113089965979128691745, −4.61553261189165064108751775504, −4.58851975239867357942218280485, −4.47825969293493488167028834391, −4.13588603613441217181197733588, −4.02805058689944312482706158185, −3.34275055849705788941246843757, −3.29025857509930432691082022509, −3.13925427812280745213278525167, −3.11158346779087527961952163799, −2.84585299933436266808038559654, −2.65658565179589186960588173441, −2.29253823753260603891475996425, −2.28667686361663449145050828473, −1.61548851523429077976484204644, −1.50314050946205367279436635983, −1.17628445191901622172309120096, −0.977826811324644584525613073324, −0.70029963210473911999286094510, −0.25551544004191191850519416008,
0.25551544004191191850519416008, 0.70029963210473911999286094510, 0.977826811324644584525613073324, 1.17628445191901622172309120096, 1.50314050946205367279436635983, 1.61548851523429077976484204644, 2.28667686361663449145050828473, 2.29253823753260603891475996425, 2.65658565179589186960588173441, 2.84585299933436266808038559654, 3.11158346779087527961952163799, 3.13925427812280745213278525167, 3.29025857509930432691082022509, 3.34275055849705788941246843757, 4.02805058689944312482706158185, 4.13588603613441217181197733588, 4.47825969293493488167028834391, 4.58851975239867357942218280485, 4.61553261189165064108751775504, 4.87397685113089965979128691745, 4.91138809342765049035436810025, 5.23373329469267946722818597361, 5.53812911921192988482297313241, 5.60189006570404819953216229041, 5.95052219106961059175235023150