| L(s) = 1 | + 2·7-s − 2·11-s − 13-s + 4·17-s − 19-s + 6·23-s + 9·29-s + 4·31-s − 3·37-s + 7·41-s − 4·43-s − 9·47-s − 3·49-s + 53-s − 6·59-s − 6·61-s + 13·67-s − 11·71-s − 8·73-s − 4·77-s − 79-s − 6·89-s − 2·91-s − 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.603·11-s − 0.277·13-s + 0.970·17-s − 0.229·19-s + 1.25·23-s + 1.67·29-s + 0.718·31-s − 0.493·37-s + 1.09·41-s − 0.609·43-s − 1.31·47-s − 3/7·49-s + 0.137·53-s − 0.781·59-s − 0.768·61-s + 1.58·67-s − 1.30·71-s − 0.936·73-s − 0.455·77-s − 0.112·79-s − 0.635·89-s − 0.209·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 11 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.05406669877671, −13.66470002960969, −13.06724433249419, −12.54795326770164, −12.14873892708612, −11.62997446505431, −11.04901594163337, −10.71215539713725, −10.02786020224755, −9.795236377297179, −9.018040123150885, −8.459043042798886, −8.075259119561896, −7.639488299981660, −6.967706427248948, −6.536638749757014, −5.792980285473701, −5.288599844290813, −4.667049842763245, −4.494553458926624, −3.429422453245654, −2.951238571332774, −2.425434300449562, −1.477934664932554, −1.040645609215248, 0,
1.040645609215248, 1.477934664932554, 2.425434300449562, 2.951238571332774, 3.429422453245654, 4.494553458926624, 4.667049842763245, 5.288599844290813, 5.792980285473701, 6.536638749757014, 6.967706427248948, 7.639488299981660, 8.075259119561896, 8.459043042798886, 9.018040123150885, 9.795236377297179, 10.02786020224755, 10.71215539713725, 11.04901594163337, 11.62997446505431, 12.14873892708612, 12.54795326770164, 13.06724433249419, 13.66470002960969, 14.05406669877671
