| L(s) = 1 | + 7-s + 2·11-s + 13-s + 17-s − 4·19-s + 7·23-s − 29-s − 3·31-s − 6·37-s + 3·41-s + 43-s − 12·47-s + 49-s − 11·53-s − 3·59-s + 5·61-s + 12·67-s + 4·71-s − 14·73-s + 2·77-s + 2·79-s − 3·83-s − 10·89-s + 91-s − 10·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.603·11-s + 0.277·13-s + 0.242·17-s − 0.917·19-s + 1.45·23-s − 0.185·29-s − 0.538·31-s − 0.986·37-s + 0.468·41-s + 0.152·43-s − 1.75·47-s + 1/7·49-s − 1.51·53-s − 0.390·59-s + 0.640·61-s + 1.46·67-s + 0.474·71-s − 1.63·73-s + 0.227·77-s + 0.225·79-s − 0.329·83-s − 1.05·89-s + 0.104·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−15.63965801402400, −14.96109560735207, −14.57783470635186, −14.16808379273261, −13.46421959417511, −12.77147151271691, −12.63917936464909, −11.72337934529166, −11.27703027822991, −10.84700646636074, −10.25288901680508, −9.466372456621276, −9.102129281171497, −8.400631445268097, −8.006495082771663, −7.130936870148273, −6.730926662280253, −6.096650732585082, −5.322064016530056, −4.819153892150578, −4.078494306006654, −3.451531636675497, −2.724126235555308, −1.773410034345060, −1.196565508886405, 0,
1.196565508886405, 1.773410034345060, 2.724126235555308, 3.451531636675497, 4.078494306006654, 4.819153892150578, 5.322064016530056, 6.096650732585082, 6.730926662280253, 7.130936870148273, 8.006495082771663, 8.400631445268097, 9.102129281171497, 9.466372456621276, 10.25288901680508, 10.84700646636074, 11.27703027822991, 11.72337934529166, 12.63917936464909, 12.77147151271691, 13.46421959417511, 14.16808379273261, 14.57783470635186, 14.96109560735207, 15.63965801402400
