| L(s) = 1 | − 5·7-s − 3·11-s + 4·13-s + 17-s − 3·19-s − 8·23-s − 9·29-s + 10·31-s + 5·37-s − 9·41-s − 6·43-s − 9·47-s + 18·49-s − 7·53-s − 2·59-s − 8·61-s + 2·67-s − 5·73-s + 15·77-s − 14·79-s − 4·83-s − 10·89-s − 20·91-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.88·7-s − 0.904·11-s + 1.10·13-s + 0.242·17-s − 0.688·19-s − 1.66·23-s − 1.67·29-s + 1.79·31-s + 0.821·37-s − 1.40·41-s − 0.914·43-s − 1.31·47-s + 18/7·49-s − 0.961·53-s − 0.260·59-s − 1.02·61-s + 0.244·67-s − 0.585·73-s + 1.70·77-s − 1.57·79-s − 0.439·83-s − 1.05·89-s − 2.09·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−14.96151815930749, −14.13035523492192, −13.55683144423230, −13.33461468993566, −12.82665962611885, −12.45698686850008, −11.75733386059284, −11.29321488393786, −10.58990876725008, −10.00170823381998, −9.916450772928227, −9.271202808350740, −8.504589416879272, −8.183378593703087, −7.565701200878200, −6.775484531794112, −6.369769583897110, −5.954894751826537, −5.479380298477288, −4.514152901139024, −3.984326305041576, −3.285293668537536, −2.986194491003932, −2.136805551462322, −1.348489054602054, 0, 0,
1.348489054602054, 2.136805551462322, 2.986194491003932, 3.285293668537536, 3.984326305041576, 4.514152901139024, 5.479380298477288, 5.954894751826537, 6.369769583897110, 6.775484531794112, 7.565701200878200, 8.183378593703087, 8.504589416879272, 9.271202808350740, 9.916450772928227, 10.00170823381998, 10.58990876725008, 11.29321488393786, 11.75733386059284, 12.45698686850008, 12.82665962611885, 13.33461468993566, 13.55683144423230, 14.13035523492192, 14.96151815930749
