| L(s) = 1 | − 2·4-s + 4·11-s + 3·16-s + 20·19-s + 8·31-s + 8·41-s − 8·44-s − 2·49-s − 20·59-s − 12·61-s − 4·64-s + 8·71-s − 40·76-s + 2·81-s − 40·89-s − 12·101-s + 40·109-s + 10·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 4-s + 1.20·11-s + 3/4·16-s + 4.58·19-s + 1.43·31-s + 1.24·41-s − 1.20·44-s − 2/7·49-s − 2.60·59-s − 1.53·61-s − 1/2·64-s + 0.949·71-s − 4.58·76-s + 2/9·81-s − 4.23·89-s − 1.19·101-s + 3.83·109-s + 0.909·121-s − 1.43·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.126647908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.126647908\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 718 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 358 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 + 20 T^{2} + 1558 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 1718 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 9238 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 6 T + 126 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 100 T^{2} + 8598 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_4$ | \( ( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 8038 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 14878 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 220 T^{2} + 25798 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.83791708322714542717812600052, −5.78571916836636751498554114074, −5.59050799345235651764144532466, −5.52189541179605798737979062990, −5.23374207378731136532822275554, −4.93848841914833130457038765595, −4.79218568564416487079530247264, −4.43645495969863769291326520343, −4.41832124569640023645857894056, −4.32906079563726233520875573296, −4.19052650269960394029648910069, −3.58791658955398145514465800048, −3.40320638655436539252217643596, −3.30437871273578586488741997646, −3.18839589877662855690976497654, −3.08925172179953132045572932829, −2.64773738519335148180714783071, −2.59301432167620350666415976450, −1.94986555002684702607324266923, −1.74718992675156946781785819827, −1.53257578986277262814719196978, −1.11449608894753470508524368225, −1.05011150078291730518513109930, −0.68787314258593422075043271021, −0.42917313236932262962014904527,
0.42917313236932262962014904527, 0.68787314258593422075043271021, 1.05011150078291730518513109930, 1.11449608894753470508524368225, 1.53257578986277262814719196978, 1.74718992675156946781785819827, 1.94986555002684702607324266923, 2.59301432167620350666415976450, 2.64773738519335148180714783071, 3.08925172179953132045572932829, 3.18839589877662855690976497654, 3.30437871273578586488741997646, 3.40320638655436539252217643596, 3.58791658955398145514465800048, 4.19052650269960394029648910069, 4.32906079563726233520875573296, 4.41832124569640023645857894056, 4.43645495969863769291326520343, 4.79218568564416487079530247264, 4.93848841914833130457038765595, 5.23374207378731136532822275554, 5.52189541179605798737979062990, 5.59050799345235651764144532466, 5.78571916836636751498554114074, 5.83791708322714542717812600052
