L(s) = 1 | − 3-s + 7-s + 9-s − 6·13-s + 2·17-s + 4·19-s − 21-s − 4·23-s − 27-s + 6·29-s − 6·37-s + 6·39-s − 2·41-s + 4·43-s − 8·47-s + 49-s − 2·51-s + 2·53-s − 4·57-s − 12·59-s + 6·61-s + 63-s + 4·67-s + 4·69-s − 12·71-s − 10·73-s − 8·79-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.66·13-s + 0.485·17-s + 0.917·19-s − 0.218·21-s − 0.834·23-s − 0.192·27-s + 1.11·29-s − 0.986·37-s + 0.960·39-s − 0.312·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.280·51-s + 0.274·53-s − 0.529·57-s − 1.56·59-s + 0.768·61-s + 0.125·63-s + 0.488·67-s + 0.481·69-s − 1.42·71-s − 1.17·73-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.83875801147440087252856722382, −7.37899047155437318008010679185, −6.60305092274033638688353303836, −5.72011333055152673676807539761, −5.02849599277547849983852859387, −4.50277956296689553328345113851, −3.37319339933434276858667925102, −2.41644630112449199782806874522, −1.33543609559105946148170448061, 0,
1.33543609559105946148170448061, 2.41644630112449199782806874522, 3.37319339933434276858667925102, 4.50277956296689553328345113851, 5.02849599277547849983852859387, 5.72011333055152673676807539761, 6.60305092274033638688353303836, 7.37899047155437318008010679185, 7.83875801147440087252856722382