| L(s) = 1 | − 2·9-s − 4·11-s − 8·29-s − 8·41-s + 12·49-s − 16·59-s + 24·61-s − 32·71-s − 16·79-s + 3·81-s − 8·89-s + 8·99-s + 8·101-s − 24·109-s + 10·121-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 + ⋯ |
| L(s) = 1 | − 2/3·9-s − 1.20·11-s − 1.48·29-s − 1.24·41-s + 12/7·49-s − 2.08·59-s + 3.07·61-s − 3.79·71-s − 1.80·79-s + 1/3·81-s − 0.847·89-s + 0.804·99-s + 0.796·101-s − 2.29·109-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 + 2.15·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1643431973\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1643431973\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 406 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 934 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 43 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 7030 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 2454 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 172 T^{2} + 14326 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 1734 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 188 T^{2} + 20566 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 9862 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.54910219048860197684352435399, −5.51888025160023007280194662805, −5.44132565738994496780858180988, −5.09330965946854835118247602648, −4.79693271256986144731052886903, −4.78110035757463184512024389049, −4.55615751672938530684623728323, −4.25676741192405420496699194965, −3.98739510958353822254291765185, −3.95307628680927908440448006726, −3.76656657058829491720556239149, −3.51589306789135908657470380723, −3.19844682559648948625093825351, −2.89757465107290086833001728768, −2.88681006699473797259067450507, −2.84108291590721533135369062738, −2.36449851041394597836632397846, −2.16125646349821558092883951364, −2.08560067852031210812645065282, −1.75549333129303068362535817582, −1.29074875555056155407966561076, −1.27595182399151331033556887355, −0.987126997214808523400509639736, −0.32587285147304587334269562344, −0.086081083921960640410573043664,
0.086081083921960640410573043664, 0.32587285147304587334269562344, 0.987126997214808523400509639736, 1.27595182399151331033556887355, 1.29074875555056155407966561076, 1.75549333129303068362535817582, 2.08560067852031210812645065282, 2.16125646349821558092883951364, 2.36449851041394597836632397846, 2.84108291590721533135369062738, 2.88681006699473797259067450507, 2.89757465107290086833001728768, 3.19844682559648948625093825351, 3.51589306789135908657470380723, 3.76656657058829491720556239149, 3.95307628680927908440448006726, 3.98739510958353822254291765185, 4.25676741192405420496699194965, 4.55615751672938530684623728323, 4.78110035757463184512024389049, 4.79693271256986144731052886903, 5.09330965946854835118247602648, 5.44132565738994496780858180988, 5.51888025160023007280194662805, 5.54910219048860197684352435399
