| L(s) = 1 | + 7-s + 4·11-s + 2·13-s + 19-s − 5·25-s − 8·29-s − 8·31-s − 10·37-s − 2·41-s + 4·43-s + 2·47-s + 49-s − 12·53-s + 4·59-s − 10·61-s − 8·67-s − 6·71-s + 6·73-s + 4·77-s − 16·79-s − 6·83-s + 6·89-s + 2·91-s + 2·97-s + 8·101-s − 16·103-s + 6·107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.20·11-s + 0.554·13-s + 0.229·19-s − 25-s − 1.48·29-s − 1.43·31-s − 1.64·37-s − 0.312·41-s + 0.609·43-s + 0.291·47-s + 1/7·49-s − 1.64·53-s + 0.520·59-s − 1.28·61-s − 0.977·67-s − 0.712·71-s + 0.702·73-s + 0.455·77-s − 1.80·79-s − 0.658·83-s + 0.635·89-s + 0.209·91-s + 0.203·97-s + 0.796·101-s − 1.57·103-s + 0.580·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.37619218558259988514690613209, −6.71178819028988536072349057242, −5.89311996311481961942417431929, −5.45023798652650480451160626362, −4.44561906949791432231170017805, −3.80508044494509671263432400283, −3.23031385657856586665491517874, −1.88504068018022093892908404351, −1.45746577688648834647305446161, 0,
1.45746577688648834647305446161, 1.88504068018022093892908404351, 3.23031385657856586665491517874, 3.80508044494509671263432400283, 4.44561906949791432231170017805, 5.45023798652650480451160626362, 5.89311996311481961942417431929, 6.71178819028988536072349057242, 7.37619218558259988514690613209
