| L(s) = 1 | − 4·4-s − 4·11-s + 7·16-s − 8·19-s + 24·29-s − 16·31-s + 24·41-s + 16·44-s − 16·49-s + 24·59-s + 8·61-s − 8·64-s + 24·71-s + 32·76-s − 8·79-s + 48·89-s + 24·101-s + 40·109-s − 96·116-s + 10·121-s + 64·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 2·4-s − 1.20·11-s + 7/4·16-s − 1.83·19-s + 4.45·29-s − 2.87·31-s + 3.74·41-s + 2.41·44-s − 2.28·49-s + 3.12·59-s + 1.02·61-s − 64-s + 2.84·71-s + 3.67·76-s − 0.900·79-s + 5.08·89-s + 2.38·101-s + 3.83·109-s − 8.91·116-s + 0.909·121-s + 5.74·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.120159842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.120159842\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p^{2} T^{2} + 9 T^{4} + p^{4} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 40 T^{2} + 786 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 + 28 T^{2} + 486 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 16 T^{2} + 2034 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 100 T^{2} + 5046 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 76 T^{2} + 3510 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 + 208 T^{2} + 19746 T^{4} + 208 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 4 T - 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 24 T + 310 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 196 T^{2} + 21510 T^{4} + 196 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.29954450941397601481191338375, −6.17118752559163255317975661271, −5.89970154666442475866712664427, −5.63276669298216188616001364069, −5.53948865592382108478690381767, −4.99025350767165704306472586260, −4.97321041744584758264870506325, −4.94888703527397153454117490509, −4.82278369246529712343834279660, −4.45117021973419528413405516416, −4.17696466558021251588437295365, −4.13662779833868656812486325303, −3.93924248431285223443488299745, −3.43216139333094944743751971899, −3.41618893920323341835173071046, −3.23841665638343542875590026052, −2.77922288601141837442465655919, −2.46914966941170226930684524310, −2.26892607367382261175162109379, −2.01204017532999959365171694095, −2.00205261363848469319925502520, −1.17448457215521381063470044501, −0.78534287314208331553003887397, −0.60932291934523407899933154577, −0.44603991102820569650947037776,
0.44603991102820569650947037776, 0.60932291934523407899933154577, 0.78534287314208331553003887397, 1.17448457215521381063470044501, 2.00205261363848469319925502520, 2.01204017532999959365171694095, 2.26892607367382261175162109379, 2.46914966941170226930684524310, 2.77922288601141837442465655919, 3.23841665638343542875590026052, 3.41618893920323341835173071046, 3.43216139333094944743751971899, 3.93924248431285223443488299745, 4.13662779833868656812486325303, 4.17696466558021251588437295365, 4.45117021973419528413405516416, 4.82278369246529712343834279660, 4.94888703527397153454117490509, 4.97321041744584758264870506325, 4.99025350767165704306472586260, 5.53948865592382108478690381767, 5.63276669298216188616001364069, 5.89970154666442475866712664427, 6.17118752559163255317975661271, 6.29954450941397601481191338375
