L(s) = 1 | + 3-s + 5-s − 2·7-s + 9-s − 2·13-s + 15-s + 4·17-s − 19-s − 2·21-s − 4·23-s + 25-s + 27-s − 6·29-s − 10·31-s − 2·35-s − 10·37-s − 2·39-s − 4·43-s + 45-s − 4·47-s − 3·49-s + 4·51-s + 10·53-s − 57-s + 6·59-s + 10·61-s − 2·63-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s + 0.970·17-s − 0.229·19-s − 0.436·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.79·31-s − 0.338·35-s − 1.64·37-s − 0.320·39-s − 0.609·43-s + 0.149·45-s − 0.583·47-s − 3/7·49-s + 0.560·51-s + 1.37·53-s − 0.132·57-s + 0.781·59-s + 1.28·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−7.957744906064071969242601726238, −7.22433433977763669397425818816, −6.65966705644225589840552614789, −5.65175441790090956548082692202, −5.19429437945824956018065244009, −3.88575745120611956452812234676, −3.43836804427240393836727470481, −2.40137790952123960259977848438, −1.61230978864933593298883413951, 0,
1.61230978864933593298883413951, 2.40137790952123960259977848438, 3.43836804427240393836727470481, 3.88575745120611956452812234676, 5.19429437945824956018065244009, 5.65175441790090956548082692202, 6.65966705644225589840552614789, 7.22433433977763669397425818816, 7.957744906064071969242601726238