| L(s) = 1 | − 3·3-s − 2·7-s + 6·9-s − 13-s + 6·21-s + 23-s − 9·27-s − 3·29-s − 3·31-s + 8·37-s + 3·39-s + 3·41-s − 2·43-s − 11·47-s − 3·49-s + 14·53-s + 8·59-s − 4·61-s − 12·63-s − 4·67-s − 3·69-s − 7·71-s + 9·73-s + 9·81-s + 4·83-s + 9·87-s − 2·89-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.755·7-s + 2·9-s − 0.277·13-s + 1.30·21-s + 0.208·23-s − 1.73·27-s − 0.557·29-s − 0.538·31-s + 1.31·37-s + 0.480·39-s + 0.468·41-s − 0.304·43-s − 1.60·47-s − 3/7·49-s + 1.92·53-s + 1.04·59-s − 0.512·61-s − 1.51·63-s − 0.488·67-s − 0.361·69-s − 0.830·71-s + 1.05·73-s + 81-s + 0.439·83-s + 0.964·87-s − 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5770959180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5770959180\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−7.43375077985563887978169616802, −6.84375350435015559761943384325, −6.31797398354246854766794516230, −5.67904551637868855962856681300, −5.14555431991860518244362280094, −4.37586317640795451613096196834, −3.65179626455037376248954783203, −2.58623822926810723667775573674, −1.42273224465244581820044878115, −0.42234562898825569737865036719,
0.42234562898825569737865036719, 1.42273224465244581820044878115, 2.58623822926810723667775573674, 3.65179626455037376248954783203, 4.37586317640795451613096196834, 5.14555431991860518244362280094, 5.67904551637868855962856681300, 6.31797398354246854766794516230, 6.84375350435015559761943384325, 7.43375077985563887978169616802
