| L(s) = 1 | − 5-s + 3·7-s + 2·11-s − 5·13-s − 8·17-s − 19-s − 6·23-s + 25-s + 2·29-s − 3·35-s + 5·37-s − 10·41-s − 4·43-s − 4·47-s + 2·49-s − 2·53-s − 2·55-s + 8·59-s + 7·61-s + 5·65-s + 9·67-s − 2·71-s − 5·73-s + 6·77-s + 3·79-s − 6·83-s + 8·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.13·7-s + 0.603·11-s − 1.38·13-s − 1.94·17-s − 0.229·19-s − 1.25·23-s + 1/5·25-s + 0.371·29-s − 0.507·35-s + 0.821·37-s − 1.56·41-s − 0.609·43-s − 0.583·47-s + 2/7·49-s − 0.274·53-s − 0.269·55-s + 1.04·59-s + 0.896·61-s + 0.620·65-s + 1.09·67-s − 0.237·71-s − 0.585·73-s + 0.683·77-s + 0.337·79-s − 0.658·83-s + 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.430790463940321537161364876409, −8.137982488844694723234284228299, −7.06111792169878582212494238113, −6.56267266288729534855302482455, −5.30117738171375778751901067843, −4.56802275334837492715788581598, −4.00540954505899980133036950237, −2.55364902597058142485713210330, −1.72341327862395054433755879505, 0,
1.72341327862395054433755879505, 2.55364902597058142485713210330, 4.00540954505899980133036950237, 4.56802275334837492715788581598, 5.30117738171375778751901067843, 6.56267266288729534855302482455, 7.06111792169878582212494238113, 8.137982488844694723234284228299, 8.430790463940321537161364876409
