| L(s) = 1 | − 4·11-s + 4·19-s + 8·29-s − 2·31-s − 22·41-s + 10·49-s + 18·59-s − 10·61-s − 2·71-s − 2·79-s + 24·89-s − 46·101-s − 10·109-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 21·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 0.917·19-s + 1.48·29-s − 0.359·31-s − 3.43·41-s + 10/7·49-s + 2.34·59-s − 1.28·61-s − 0.237·71-s − 0.225·79-s + 2.54·89-s − 4.57·101-s − 0.957·109-s + 0.909·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.61·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07571572654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07571572654\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 21 T^{2} + 412 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 9 T^{2} + 16 p T^{4} + 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 - 7 T^{2} + 164 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 + T + 26 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 63 T^{2} + 2824 T^{4} - 63 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 11 T + 76 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 59 T^{2} + 1632 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 75 T^{2} + 2888 T^{4} - 75 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 9 T + 102 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 5 T + 92 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 15 T^{2} - 4052 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + T + 106 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + T + 122 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 13598 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 90 T^{2} + 18523 T^{4} - 90 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.27033683230734376998814501096, −5.26800175174697970881374834530, −5.18339359050377180582197900955, −4.78757674603825134560512400879, −4.66356102131741425528100012508, −4.46895964118218847022530733027, −4.46892130289290669824317320818, −3.88862484243525947374620832133, −3.85922608537989797535136702841, −3.78485921277467481779337734420, −3.53216919417839205666364018707, −3.34829178346296955640430290361, −2.98827203260322351614480199681, −2.88253730930274350388828723805, −2.81980473409049221611434100691, −2.45247854847071643418254244601, −2.39604100089448707654983430976, −2.06158247462042068926037450949, −1.88742219541146408923756497289, −1.54093281423628079991893726174, −1.35627885894512844696235767803, −1.03027403574657027502551974177, −0.948321312291163334207701466377, −0.44732252933028528883935978831, −0.03727014420918062378200552328,
0.03727014420918062378200552328, 0.44732252933028528883935978831, 0.948321312291163334207701466377, 1.03027403574657027502551974177, 1.35627885894512844696235767803, 1.54093281423628079991893726174, 1.88742219541146408923756497289, 2.06158247462042068926037450949, 2.39604100089448707654983430976, 2.45247854847071643418254244601, 2.81980473409049221611434100691, 2.88253730930274350388828723805, 2.98827203260322351614480199681, 3.34829178346296955640430290361, 3.53216919417839205666364018707, 3.78485921277467481779337734420, 3.85922608537989797535136702841, 3.88862484243525947374620832133, 4.46892130289290669824317320818, 4.46895964118218847022530733027, 4.66356102131741425528100012508, 4.78757674603825134560512400879, 5.18339359050377180582197900955, 5.26800175174697970881374834530, 5.27033683230734376998814501096
