| L(s) = 1 | − 3-s − 3·5-s − 3·7-s + 9-s − 11-s + 2·13-s + 3·15-s − 5·17-s + 3·21-s − 4·23-s + 4·25-s − 27-s + 6·29-s + 2·31-s + 33-s + 9·35-s − 8·37-s − 2·39-s + 8·41-s + 13·43-s − 3·45-s + 13·47-s + 2·49-s + 5·51-s + 6·53-s + 3·55-s − 4·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 1.13·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.774·15-s − 1.21·17-s + 0.654·21-s − 0.834·23-s + 4/5·25-s − 0.192·27-s + 1.11·29-s + 0.359·31-s + 0.174·33-s + 1.52·35-s − 1.31·37-s − 0.320·39-s + 1.24·41-s + 1.98·43-s − 0.447·45-s + 1.89·47-s + 2/7·49-s + 0.700·51-s + 0.824·53-s + 0.404·55-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 13 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.48722452079366411338580209149, −6.61817243450412613583037515362, −6.23594922502770403826571343940, −5.41339600489132443742989782718, −4.29415537117189900002627510921, −4.11309802622817372680738239516, −3.17926980688467316862081276218, −2.34548339342499975775604881106, −0.835191651314275551479017177030, 0,
0.835191651314275551479017177030, 2.34548339342499975775604881106, 3.17926980688467316862081276218, 4.11309802622817372680738239516, 4.29415537117189900002627510921, 5.41339600489132443742989782718, 6.23594922502770403826571343940, 6.61817243450412613583037515362, 7.48722452079366411338580209149
