| L(s) = 1 | − 3-s + 5-s − 2·9-s + 2·11-s − 4·13-s − 15-s + 6·19-s − 3·23-s + 25-s + 5·27-s − 3·29-s − 2·33-s − 12·37-s + 4·39-s + 7·41-s + 9·43-s − 2·45-s − 6·53-s + 2·55-s − 6·57-s − 10·59-s − 5·61-s − 4·65-s − 11·67-s + 3·69-s + 10·71-s + 8·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 2/3·9-s + 0.603·11-s − 1.10·13-s − 0.258·15-s + 1.37·19-s − 0.625·23-s + 1/5·25-s + 0.962·27-s − 0.557·29-s − 0.348·33-s − 1.97·37-s + 0.640·39-s + 1.09·41-s + 1.37·43-s − 0.298·45-s − 0.824·53-s + 0.269·55-s − 0.794·57-s − 1.30·59-s − 0.640·61-s − 0.496·65-s − 1.34·67-s + 0.361·69-s + 1.18·71-s + 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 17 T + p T^{2} \) |
| 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
−7.988330696642812328871810984430, −7.32999126973217434560733636642, −6.55977579557836438808464395431, −5.74927040774399969149906569707, −5.29208014638074091425863495821, −4.43472520854776928117273483310, −3.34806454923890819346910354384, −2.49118771006450064224323948705, −1.35778867624689297279899334700, 0,
1.35778867624689297279899334700, 2.49118771006450064224323948705, 3.34806454923890819346910354384, 4.43472520854776928117273483310, 5.29208014638074091425863495821, 5.74927040774399969149906569707, 6.55977579557836438808464395431, 7.32999126973217434560733636642, 7.988330696642812328871810984430
