L(s) = 1 | − 2·3-s + 9-s + 5·11-s − 8·17-s + 2·19-s − 7·23-s + 4·27-s − 3·29-s − 4·31-s − 10·33-s − 37-s + 2·41-s + 3·43-s − 6·47-s + 16·51-s + 10·53-s − 4·57-s + 4·59-s + 6·61-s + 13·67-s + 14·69-s + 5·71-s − 6·73-s − 13·79-s − 11·81-s − 16·83-s + 6·87-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/3·9-s + 1.50·11-s − 1.94·17-s + 0.458·19-s − 1.45·23-s + 0.769·27-s − 0.557·29-s − 0.718·31-s − 1.74·33-s − 0.164·37-s + 0.312·41-s + 0.457·43-s − 0.875·47-s + 2.24·51-s + 1.37·53-s − 0.529·57-s + 0.520·59-s + 0.768·61-s + 1.58·67-s + 1.68·69-s + 0.593·71-s − 0.702·73-s − 1.46·79-s − 1.22·81-s − 1.75·83-s + 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9107584590\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9107584590\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−7.40156902623104112136135293517, −6.81974476302015993538098907767, −6.28030089034587164914019908907, −5.75886131497957428161104434441, −4.98682291407930254137866999637, −4.17603364925740276076305692247, −3.74997472783033928539568023178, −2.43113428855824507821465967532, −1.60022093640587338519606985992, −0.48713894443212357469380882520,
0.48713894443212357469380882520, 1.60022093640587338519606985992, 2.43113428855824507821465967532, 3.74997472783033928539568023178, 4.17603364925740276076305692247, 4.98682291407930254137866999637, 5.75886131497957428161104434441, 6.28030089034587164914019908907, 6.81974476302015993538098907767, 7.40156902623104112136135293517