| L(s) = 1 | − 4·4-s + 12·16-s + 24·19-s + 4·25-s − 12·31-s − 16·37-s + 14·49-s − 32·64-s − 96·76-s − 16·100-s + 60·103-s + 20·109-s − 20·121-s + 48·124-s + 127-s + 131-s + 137-s + 139-s + 64·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 2·4-s + 3·16-s + 5.50·19-s + 4/5·25-s − 2.15·31-s − 2.63·37-s + 2·49-s − 4·64-s − 11.0·76-s − 8/5·100-s + 5.91·103-s + 1.91·109-s − 1.81·121-s + 4.31·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.26·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.174998881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.174998881\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 - 4 T^{2} - 9 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 20 T^{2} + 279 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 12 T + 67 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 4 T^{2} - 513 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 + 68 T^{2} + 2943 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2^3$ | \( 1 - 100 T^{2} + 4959 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2^3$ | \( 1 - 172 T^{2} + 21663 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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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
−7.66314493365880711670673427078, −7.28229099466950122468372757604, −7.08268979643034831742203351763, −6.76194821437538189499250642039, −6.75084894007339945859251442541, −6.01078178046863748000684862628, −5.87564668180646150953077152587, −5.57991543760693537946969347507, −5.47946503328480549862527402453, −5.27743524326289546757876085360, −5.20216638715895349612941496002, −4.82497260472910764220009063519, −4.63618959596965044482755173777, −4.39750783513247884967071630298, −3.86378777124526290326446372242, −3.59771604547932269346130524603, −3.46423420805282078299207995876, −3.34880070376196398890156903592, −3.07048286821008253249965153798, −2.77051087259050112507234502325, −2.05995356233578782898175657840, −1.66729105590238308802836333247, −1.30672888097311823433414207385, −0.77113942865948026636929134961, −0.62250943643370950963610734478,
0.62250943643370950963610734478, 0.77113942865948026636929134961, 1.30672888097311823433414207385, 1.66729105590238308802836333247, 2.05995356233578782898175657840, 2.77051087259050112507234502325, 3.07048286821008253249965153798, 3.34880070376196398890156903592, 3.46423420805282078299207995876, 3.59771604547932269346130524603, 3.86378777124526290326446372242, 4.39750783513247884967071630298, 4.63618959596965044482755173777, 4.82497260472910764220009063519, 5.20216638715895349612941496002, 5.27743524326289546757876085360, 5.47946503328480549862527402453, 5.57991543760693537946969347507, 5.87564668180646150953077152587, 6.01078178046863748000684862628, 6.75084894007339945859251442541, 6.76194821437538189499250642039, 7.08268979643034831742203351763, 7.28229099466950122468372757604, 7.66314493365880711670673427078
