| L(s) = 1 | − 7-s − 11-s + 2·13-s + 8·17-s + 2·19-s − 23-s − 29-s − 6·31-s − 9·37-s − 43-s − 6·47-s + 49-s + 2·53-s − 6·59-s + 8·61-s − 3·67-s + 7·71-s − 16·73-s + 77-s − 79-s + 6·83-s + 14·89-s − 2·91-s − 14·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 0.301·11-s + 0.554·13-s + 1.94·17-s + 0.458·19-s − 0.208·23-s − 0.185·29-s − 1.07·31-s − 1.47·37-s − 0.152·43-s − 0.875·47-s + 1/7·49-s + 0.274·53-s − 0.781·59-s + 1.02·61-s − 0.366·67-s + 0.830·71-s − 1.87·73-s + 0.113·77-s − 0.112·79-s + 0.658·83-s + 1.48·89-s − 0.209·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.83859242318075, −14.96817327625766, −14.59587121706407, −14.02780280338668, −13.48930981653825, −12.94629476000515, −12.37324695942167, −11.92338092679208, −11.36551804393540, −10.58683281179109, −10.23796562650339, −9.645125875442794, −9.079713690921591, −8.437368646760795, −7.789961787541318, −7.369345093767845, −6.662462121149933, −5.952205720204954, −5.421205265750574, −4.959460954120137, −3.840281788366692, −3.481287612962755, −2.839151101977786, −1.801630608787072, −1.108264493975370, 0,
1.108264493975370, 1.801630608787072, 2.839151101977786, 3.481287612962755, 3.840281788366692, 4.959460954120137, 5.421205265750574, 5.952205720204954, 6.662462121149933, 7.369345093767845, 7.789961787541318, 8.437368646760795, 9.079713690921591, 9.645125875442794, 10.23796562650339, 10.58683281179109, 11.36551804393540, 11.92338092679208, 12.37324695942167, 12.94629476000515, 13.48930981653825, 14.02780280338668, 14.59587121706407, 14.96817327625766, 15.83859242318075
