L(s) = 1 | + 3-s + 4·7-s + 9-s − 4·11-s + 4·17-s + 4·21-s + 4·23-s + 27-s − 6·29-s + 4·31-s − 4·33-s − 8·37-s − 10·41-s + 4·43-s − 4·47-s + 9·49-s + 4·51-s − 12·53-s + 4·59-s + 2·61-s + 4·63-s − 4·67-s + 4·69-s − 8·73-s − 16·77-s − 12·79-s + 81-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s − 1.20·11-s + 0.970·17-s + 0.872·21-s + 0.834·23-s + 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.696·33-s − 1.31·37-s − 1.56·41-s + 0.609·43-s − 0.583·47-s + 9/7·49-s + 0.560·51-s − 1.64·53-s + 0.520·59-s + 0.256·61-s + 0.503·63-s − 0.488·67-s + 0.481·69-s − 0.936·73-s − 1.82·77-s − 1.35·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.702522910\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.702522910\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−11.66423615429283421096728190882, −10.79400555935724064711962029744, −9.939583129965683410898359877662, −8.663418930560570598737303073543, −7.985267473843071116990190639373, −7.22065862859774326573619751365, −5.49518474552018854569919656545, −4.70569070648500642379588704382, −3.16765979420003827972319820965, −1.72530323216821322522551991647,
1.72530323216821322522551991647, 3.16765979420003827972319820965, 4.70569070648500642379588704382, 5.49518474552018854569919656545, 7.22065862859774326573619751365, 7.985267473843071116990190639373, 8.663418930560570598737303073543, 9.939583129965683410898359877662, 10.79400555935724064711962029744, 11.66423615429283421096728190882