| L(s) = 1 | + 2·3-s + 9-s − 4·11-s + 4·13-s + 4·19-s + 2·23-s − 4·27-s + 2·29-s − 8·33-s − 4·37-s + 8·39-s − 2·41-s + 6·43-s − 6·47-s + 4·53-s + 8·57-s + 12·59-s + 10·61-s − 14·67-s + 4·69-s + 8·71-s + 8·73-s + 16·79-s − 11·81-s + 2·83-s + 4·87-s − 6·89-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s − 1.20·11-s + 1.10·13-s + 0.917·19-s + 0.417·23-s − 0.769·27-s + 0.371·29-s − 1.39·33-s − 0.657·37-s + 1.28·39-s − 0.312·41-s + 0.914·43-s − 0.875·47-s + 0.549·53-s + 1.05·57-s + 1.56·59-s + 1.28·61-s − 1.71·67-s + 0.481·69-s + 0.949·71-s + 0.936·73-s + 1.80·79-s − 1.22·81-s + 0.219·83-s + 0.428·87-s − 0.635·89-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
| \(L(1)\) |
\(\approx\) |
\(3.130907948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.130907948\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−7.80490559912864830789451763308, −7.20511912671126545376415987851, −6.36916259218775338867521337025, −5.51293166761659743969374174316, −5.00192751095038063419861633350, −3.89903784798328546362956718852, −3.34995357407111759567627632435, −2.68534013766286934612944284896, −1.93552250057563034818517220292, −0.788949000917484654700473342274,
0.788949000917484654700473342274, 1.93552250057563034818517220292, 2.68534013766286934612944284896, 3.34995357407111759567627632435, 3.89903784798328546362956718852, 5.00192751095038063419861633350, 5.51293166761659743969374174316, 6.36916259218775338867521337025, 7.20511912671126545376415987851, 7.80490559912864830789451763308
