| L(s) = 1 | + 2-s − 3-s − 5-s − 6-s − 8-s − 10-s + 11-s − 2·13-s + 15-s − 16-s − 3·19-s + 22-s − 7·23-s + 24-s − 2·26-s + 27-s − 16·29-s + 30-s − 2·31-s − 33-s − 11·37-s − 3·38-s + 2·39-s + 40-s + 22·41-s + 16·43-s − 7·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.447·5-s − 0.408·6-s − 0.353·8-s − 0.316·10-s + 0.301·11-s − 0.554·13-s + 0.258·15-s − 1/4·16-s − 0.688·19-s + 0.213·22-s − 1.45·23-s + 0.204·24-s − 0.392·26-s + 0.192·27-s − 2.97·29-s + 0.182·30-s − 0.359·31-s − 0.174·33-s − 1.80·37-s − 0.486·38-s + 0.320·39-s + 0.158·40-s + 3.43·41-s + 2.43·43-s − 1.03·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7746265433\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7746265433\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 3 T - 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 5 T - 22 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 11 T + 68 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
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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
−9.753454417185625800572825285905, −9.146353683229800724582652377772, −8.986722804677901127227257430761, −8.693723195080642662925728377664, −7.79346106087266620485001231542, −7.68998901340847150621717626859, −7.30966273343146199358405903726, −6.92262637012188349132149420093, −6.15990554813016074397789566380, −5.94326101734480387602690913637, −5.59649745273060555980122573302, −5.29671005939261916869100566876, −4.42344079111448365833187729963, −4.33181151033038606731622145041, −3.73272268281713611396952477324, −3.59067646456588672500919073783, −2.45322509333451396517058481957, −2.33887324136620453230471265959, −1.42972529726077457029203743100, −0.32403681845762107195321269734,
0.32403681845762107195321269734, 1.42972529726077457029203743100, 2.33887324136620453230471265959, 2.45322509333451396517058481957, 3.59067646456588672500919073783, 3.73272268281713611396952477324, 4.33181151033038606731622145041, 4.42344079111448365833187729963, 5.29671005939261916869100566876, 5.59649745273060555980122573302, 5.94326101734480387602690913637, 6.15990554813016074397789566380, 6.92262637012188349132149420093, 7.30966273343146199358405903726, 7.68998901340847150621717626859, 7.79346106087266620485001231542, 8.693723195080642662925728377664, 8.986722804677901127227257430761, 9.146353683229800724582652377772, 9.753454417185625800572825285905
