| L(s) = 1 | − 7-s − 5·11-s + 5·13-s − 5·17-s − 8·23-s − 29-s − 2·31-s + 4·37-s − 2·41-s − 4·43-s − 13·47-s + 49-s + 8·53-s − 4·59-s + 2·61-s + 8·67-s − 12·71-s + 6·73-s + 5·77-s − 11·79-s + 8·83-s + 18·89-s − 5·91-s + 5·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.50·11-s + 1.38·13-s − 1.21·17-s − 1.66·23-s − 0.185·29-s − 0.359·31-s + 0.657·37-s − 0.312·41-s − 0.609·43-s − 1.89·47-s + 1/7·49-s + 1.09·53-s − 0.520·59-s + 0.256·61-s + 0.977·67-s − 1.42·71-s + 0.702·73-s + 0.569·77-s − 1.23·79-s + 0.878·83-s + 1.90·89-s − 0.524·91-s + 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−13.77874376883988, −13.41485876489579, −13.08314917932863, −12.77610277176440, −11.94012022109722, −11.52984358408312, −11.03248900844756, −10.51306201477662, −10.15962587130885, −9.608048025551777, −8.975654006180620, −8.416165415775636, −8.116995880890712, −7.557405259575924, −6.884491359702758, −6.307686977314675, −5.941122075462727, −5.372847366322506, −4.682681719035203, −4.189059017236462, −3.462041955434723, −3.069736690875380, −2.095355172208550, −1.907496777925430, −0.7042408540963858, 0,
0.7042408540963858, 1.907496777925430, 2.095355172208550, 3.069736690875380, 3.462041955434723, 4.189059017236462, 4.682681719035203, 5.372847366322506, 5.941122075462727, 6.307686977314675, 6.884491359702758, 7.557405259575924, 8.116995880890712, 8.416165415775636, 8.975654006180620, 9.608048025551777, 10.15962587130885, 10.51306201477662, 11.03248900844756, 11.52984358408312, 11.94012022109722, 12.77610277176440, 13.08314917932863, 13.41485876489579, 13.77874376883988
