| L(s) = 1 | − 5-s + 5·7-s + 11-s + 13-s + 3·17-s − 6·19-s − 7·23-s + 25-s − 6·29-s + 2·31-s − 5·35-s + 37-s − 7·41-s − 8·43-s + 2·47-s + 18·49-s − 13·53-s − 55-s − 8·59-s − 7·61-s − 65-s − 12·67-s + 71-s − 10·73-s + 5·77-s − 3·79-s + 8·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.88·7-s + 0.301·11-s + 0.277·13-s + 0.727·17-s − 1.37·19-s − 1.45·23-s + 1/5·25-s − 1.11·29-s + 0.359·31-s − 0.845·35-s + 0.164·37-s − 1.09·41-s − 1.21·43-s + 0.291·47-s + 18/7·49-s − 1.78·53-s − 0.134·55-s − 1.04·59-s − 0.896·61-s − 0.124·65-s − 1.46·67-s + 0.118·71-s − 1.17·73-s + 0.569·77-s − 0.337·79-s + 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−7.77274098934623134106498841667, −6.68456463956284461992549819853, −5.96049492196497770192387276286, −5.23359546378970092672757751564, −4.45593366487140699005560723946, −4.07850751804359064282435449571, −3.08892471955957328530821591145, −1.85020610880961958957260450690, −1.52020192541098863611220659668, 0,
1.52020192541098863611220659668, 1.85020610880961958957260450690, 3.08892471955957328530821591145, 4.07850751804359064282435449571, 4.45593366487140699005560723946, 5.23359546378970092672757751564, 5.96049492196497770192387276286, 6.68456463956284461992549819853, 7.77274098934623134106498841667
