| L(s) = 1 | − 4·5-s + 10·25-s + 24·37-s + 16·41-s − 16·43-s − 8·47-s + 6·49-s + 24·59-s − 40·83-s − 8·89-s − 16·101-s + 40·121-s − 20·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s − 96·185-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 2·25-s + 3.94·37-s + 2.49·41-s − 2.43·43-s − 1.16·47-s + 6/7·49-s + 3.12·59-s − 4.39·83-s − 0.847·89-s − 1.59·101-s + 3.63·121-s − 1.78·125-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s − 7.05·185-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.318202087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.318202087\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 454 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1318 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 36 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^3$ | \( 1 - 1198 T^{4} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4342 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 8434 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 148 T^{2} + 11638 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 71 | $D_4\times C_2$ | \( 1 - 200 T^{2} + 19762 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 12 T^{2} - 7306 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 12 T^{2} - 16426 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.80987512022307825930822785249, −5.64097301471723458646861737861, −5.35297621408159353866983394309, −5.27106918183385538756099025285, −5.16947070326421111618961918822, −4.51084030402442142424931989491, −4.48668528273645559149631183523, −4.40850727406688185400815148652, −4.28396009961050608435603962899, −4.13651572849901313668264050420, −4.04929488918519528737747578072, −3.53397506577129661715868181113, −3.45395345898569606032165779674, −3.08556204117722952177850373967, −3.08480778395926153056354297498, −2.73089335234157962949463127822, −2.72452873323299801562513930890, −2.31144525918605813675467676842, −2.07172810671072841019129682619, −1.68937854370445925633959032688, −1.60894131982255124200559429052, −0.860183592958868631667176438513, −0.854037938871488315644663995115, −0.800271339068385883547817785498, −0.18466653628011852168196656936,
0.18466653628011852168196656936, 0.800271339068385883547817785498, 0.854037938871488315644663995115, 0.860183592958868631667176438513, 1.60894131982255124200559429052, 1.68937854370445925633959032688, 2.07172810671072841019129682619, 2.31144525918605813675467676842, 2.72452873323299801562513930890, 2.73089335234157962949463127822, 3.08480778395926153056354297498, 3.08556204117722952177850373967, 3.45395345898569606032165779674, 3.53397506577129661715868181113, 4.04929488918519528737747578072, 4.13651572849901313668264050420, 4.28396009961050608435603962899, 4.40850727406688185400815148652, 4.48668528273645559149631183523, 4.51084030402442142424931989491, 5.16947070326421111618961918822, 5.27106918183385538756099025285, 5.35297621408159353866983394309, 5.64097301471723458646861737861, 5.80987512022307825930822785249
