| L(s) = 1 | − 2·3-s + 2·5-s + 3·9-s + 4·11-s − 4·15-s + 3·25-s − 4·27-s − 8·33-s − 4·37-s + 6·45-s + 2·49-s − 4·53-s + 8·55-s + 8·59-s + 8·67-s − 16·71-s − 6·75-s + 5·81-s + 4·89-s − 12·97-s + 12·99-s + 16·103-s + 8·111-s − 28·113-s + 5·121-s + 4·125-s + 127-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 9-s + 1.20·11-s − 1.03·15-s + 3/5·25-s − 0.769·27-s − 1.39·33-s − 0.657·37-s + 0.894·45-s + 2/7·49-s − 0.549·53-s + 1.07·55-s + 1.04·59-s + 0.977·67-s − 1.89·71-s − 0.692·75-s + 5/9·81-s + 0.423·89-s − 1.21·97-s + 1.20·99-s + 1.57·103-s + 0.759·111-s − 2.63·113-s + 5/11·121-s + 0.357·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.787139631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.787139631\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.69772309123989108220606203680, −7.24286157616606608479018143502, −6.78551843031724643856071699061, −6.49741853669412748069238506316, −6.17597213369972722055652239886, −5.64747469702184415761761201344, −5.31182846678436450493419044437, −4.93163920411726928209504798837, −4.22136173162431401656857580472, −4.01790463702023132038220328928, −3.28401677559658237098373233655, −2.64729285031370884332041299776, −1.83312502448824348368144578655, −1.44894769090136273852843086875, −0.63268977118096701538118248868,
0.63268977118096701538118248868, 1.44894769090136273852843086875, 1.83312502448824348368144578655, 2.64729285031370884332041299776, 3.28401677559658237098373233655, 4.01790463702023132038220328928, 4.22136173162431401656857580472, 4.93163920411726928209504798837, 5.31182846678436450493419044437, 5.64747469702184415761761201344, 6.17597213369972722055652239886, 6.49741853669412748069238506316, 6.78551843031724643856071699061, 7.24286157616606608479018143502, 7.69772309123989108220606203680
