| L(s) = 1 | − 3-s + 9-s − 6·11-s − 2·13-s + 2·19-s − 4·23-s − 27-s + 2·29-s − 8·31-s + 6·33-s + 8·37-s + 2·39-s − 2·41-s − 43-s + 12·47-s − 7·49-s − 8·53-s − 2·57-s − 2·59-s − 6·61-s − 4·67-s + 4·69-s − 12·71-s + 81-s + 12·83-s − 2·87-s + 6·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.80·11-s − 0.554·13-s + 0.458·19-s − 0.834·23-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 1.04·33-s + 1.31·37-s + 0.320·39-s − 0.312·41-s − 0.152·43-s + 1.75·47-s − 49-s − 1.09·53-s − 0.264·57-s − 0.260·59-s − 0.768·61-s − 0.488·67-s + 0.481·69-s − 1.42·71-s + 1/9·81-s + 1.31·83-s − 0.214·87-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4368757324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4368757324\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.59314637823498, −13.83541381667747, −13.43242774590558, −12.91283131155943, −12.38343049699246, −12.08622457332091, −11.21870765417114, −11.00131298153330, −10.27510760798150, −10.04576122098121, −9.363411217965190, −8.792643748563307, −7.956423989426050, −7.604946384528247, −7.300167230021830, −6.313624116554137, −5.945554535359522, −5.255279413020455, −4.913660672466673, −4.261538508272923, −3.452019148454616, −2.730445778350844, −2.198943720863848, −1.317860449767635, −0.2408782833768973,
0.2408782833768973, 1.317860449767635, 2.198943720863848, 2.730445778350844, 3.452019148454616, 4.261538508272923, 4.913660672466673, 5.255279413020455, 5.945554535359522, 6.313624116554137, 7.300167230021830, 7.604946384528247, 7.956423989426050, 8.792643748563307, 9.363411217965190, 10.04576122098121, 10.27510760798150, 11.00131298153330, 11.21870765417114, 12.08622457332091, 12.38343049699246, 12.91283131155943, 13.43242774590558, 13.83541381667747, 14.59314637823498
