| L(s) = 1 | + 2·3-s − 5-s + 2·7-s + 9-s + 6·13-s − 2·15-s + 2·17-s − 19-s + 4·21-s − 2·23-s + 25-s − 4·27-s − 2·29-s + 4·31-s − 2·35-s − 10·37-s + 12·39-s − 10·41-s + 6·43-s − 45-s − 6·47-s − 3·49-s + 4·51-s + 6·53-s − 2·57-s − 4·59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.66·13-s − 0.516·15-s + 0.485·17-s − 0.229·19-s + 0.872·21-s − 0.417·23-s + 1/5·25-s − 0.769·27-s − 0.371·29-s + 0.718·31-s − 0.338·35-s − 1.64·37-s + 1.92·39-s − 1.56·41-s + 0.914·43-s − 0.149·45-s − 0.875·47-s − 3/7·49-s + 0.560·51-s + 0.824·53-s − 0.264·57-s − 0.520·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.961072164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.961072164\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−11.32508456896308415489084259774, −10.48496671408108499284220324727, −9.272396612094169685912601224152, −8.294000637754294355036016345474, −8.127953611229209522722729832311, −6.78152084585319822916934144821, −5.46860793773004683410757765520, −4.05957201430371047965905503497, −3.21566207427278766631032003624, −1.68036196072528569368485026712,
1.68036196072528569368485026712, 3.21566207427278766631032003624, 4.05957201430371047965905503497, 5.46860793773004683410757765520, 6.78152084585319822916934144821, 8.127953611229209522722729832311, 8.294000637754294355036016345474, 9.272396612094169685912601224152, 10.48496671408108499284220324727, 11.32508456896308415489084259774
