| L(s) = 1 | − 2·5-s + 10·7-s − 8·11-s − 5·13-s − 8·19-s + 6·23-s + 5·25-s + 8·29-s + 2·31-s − 20·35-s + 14·37-s − 11·43-s − 10·47-s + 61·49-s + 6·53-s + 16·55-s − 8·59-s + 61-s + 10·65-s + 5·67-s − 6·71-s − 73-s − 80·77-s − 13·79-s − 8·83-s − 12·89-s − 50·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 3.77·7-s − 2.41·11-s − 1.38·13-s − 1.83·19-s + 1.25·23-s + 25-s + 1.48·29-s + 0.359·31-s − 3.38·35-s + 2.30·37-s − 1.67·43-s − 1.45·47-s + 61/7·49-s + 0.824·53-s + 2.15·55-s − 1.04·59-s + 0.128·61-s + 1.24·65-s + 0.610·67-s − 0.712·71-s − 0.117·73-s − 9.11·77-s − 1.46·79-s − 0.878·83-s − 1.27·89-s − 5.24·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.240091851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.240091851\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 10 T + 53 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + T - 72 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.706352139263630936044012110245, −8.343676052850351411112881621887, −8.144241001324091353134128451947, −8.136805667153587877195395437590, −7.59322712672205447095886873809, −7.30712905044490937376951120611, −7.06707039432137499667419833894, −6.35586218098762543330109394937, −5.73366441506948604091137256453, −5.16114020840914364488774191536, −5.10547977400353283724261130863, −4.59282919297359213102333104112, −4.48311017739948900250897140516, −4.36692886954130056017159836809, −3.21061298426581242274874067147, −2.58112772858305102947161695747, −2.49648583472268884474435854334, −1.87338182415055904577700401789, −1.28703331155673107417395809003, −0.50021097979098971548336206999,
0.50021097979098971548336206999, 1.28703331155673107417395809003, 1.87338182415055904577700401789, 2.49648583472268884474435854334, 2.58112772858305102947161695747, 3.21061298426581242274874067147, 4.36692886954130056017159836809, 4.48311017739948900250897140516, 4.59282919297359213102333104112, 5.10547977400353283724261130863, 5.16114020840914364488774191536, 5.73366441506948604091137256453, 6.35586218098762543330109394937, 7.06707039432137499667419833894, 7.30712905044490937376951120611, 7.59322712672205447095886873809, 8.136805667153587877195395437590, 8.144241001324091353134128451947, 8.343676052850351411112881621887, 8.706352139263630936044012110245
