| L(s) = 1 | − 2·3-s + 7-s + 9-s + 11-s − 4·13-s + 6·19-s − 2·21-s − 3·23-s + 4·27-s − 3·29-s − 2·33-s − 9·37-s + 8·39-s + 2·41-s − 9·43-s + 6·47-s + 49-s − 6·53-s − 12·57-s + 8·59-s − 10·61-s + 63-s + 67-s + 6·69-s − 7·71-s + 2·73-s + 77-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 1.37·19-s − 0.436·21-s − 0.625·23-s + 0.769·27-s − 0.557·29-s − 0.348·33-s − 1.47·37-s + 1.28·39-s + 0.312·41-s − 1.37·43-s + 0.875·47-s + 1/7·49-s − 0.824·53-s − 1.58·57-s + 1.04·59-s − 1.28·61-s + 0.125·63-s + 0.122·67-s + 0.722·69-s − 0.830·71-s + 0.234·73-s + 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−9.328240328959893836820488605573, −8.291638824354553749832147975540, −7.36093716429216483962761853710, −6.71609133607624229944036178184, −5.60555227321757038670706914438, −5.20829029475748356758143176483, −4.22360783809719513353208535534, −2.94349967687063095785099679694, −1.51296624879668184664231800532, 0,
1.51296624879668184664231800532, 2.94349967687063095785099679694, 4.22360783809719513353208535534, 5.20829029475748356758143176483, 5.60555227321757038670706914438, 6.71609133607624229944036178184, 7.36093716429216483962761853710, 8.291638824354553749832147975540, 9.328240328959893836820488605573
