| L(s) = 1 | − 2·3-s − 3·7-s + 9-s + 6·11-s − 4·13-s − 3·17-s − 6·19-s + 6·21-s + 4·27-s + 3·29-s + 5·31-s − 12·33-s − 3·37-s + 8·39-s + 9·41-s − 12·43-s − 6·47-s + 2·49-s + 6·51-s − 3·53-s + 12·57-s + 9·59-s + 6·61-s − 3·63-s + 9·67-s − 9·71-s + 2·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.13·7-s + 1/3·9-s + 1.80·11-s − 1.10·13-s − 0.727·17-s − 1.37·19-s + 1.30·21-s + 0.769·27-s + 0.557·29-s + 0.898·31-s − 2.08·33-s − 0.493·37-s + 1.28·39-s + 1.40·41-s − 1.82·43-s − 0.875·47-s + 2/7·49-s + 0.840·51-s − 0.412·53-s + 1.58·57-s + 1.17·59-s + 0.768·61-s − 0.377·63-s + 1.09·67-s − 1.06·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.18077538121525, −12.51285561167952, −12.32616042071423, −11.89550660974162, −11.36476060997803, −11.09995521234511, −10.37736505927218, −10.01264964940263, −9.569643784713013, −9.142202012489953, −8.490371274578362, −8.215909499827228, −7.159015280405445, −6.793437814707687, −6.493825381117262, −6.231390972305955, −5.626645122651649, −4.803571344351784, −4.643949767818988, −3.920891541345971, −3.463909509505857, −2.675831378652761, −2.155613244127206, −1.330702406153460, −0.5830848868546298, 0,
0.5830848868546298, 1.330702406153460, 2.155613244127206, 2.675831378652761, 3.463909509505857, 3.920891541345971, 4.643949767818988, 4.803571344351784, 5.626645122651649, 6.231390972305955, 6.493825381117262, 6.793437814707687, 7.159015280405445, 8.215909499827228, 8.490371274578362, 9.142202012489953, 9.569643784713013, 10.01264964940263, 10.37736505927218, 11.09995521234511, 11.36476060997803, 11.89550660974162, 12.32616042071423, 12.51285561167952, 13.18077538121525
