| L(s) = 1 | + 7-s − 4·11-s − 3·13-s − 7·17-s + 6·19-s + 9·23-s + 3·29-s + 7·31-s − 10·37-s − 41-s − 13·43-s − 2·47-s + 49-s + 53-s + 11·59-s + 13·61-s − 8·71-s + 8·73-s − 4·77-s − 4·79-s + 7·83-s − 14·89-s − 3·91-s + 8·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.20·11-s − 0.832·13-s − 1.69·17-s + 1.37·19-s + 1.87·23-s + 0.557·29-s + 1.25·31-s − 1.64·37-s − 0.156·41-s − 1.98·43-s − 0.291·47-s + 1/7·49-s + 0.137·53-s + 1.43·59-s + 1.66·61-s − 0.949·71-s + 0.936·73-s − 0.455·77-s − 0.450·79-s + 0.768·83-s − 1.48·89-s − 0.314·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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
| \(L(1)\) |
\(\approx\) |
\(1.592747519\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.592747519\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.23498116621228, −15.06843626882589, −14.24367038245539, −13.64734043138642, −13.25117331328908, −12.83171951157290, −12.00591680967286, −11.57431683155205, −11.09442277580184, −10.35015401203907, −10.05165247327406, −9.319356567538550, −8.594795071817180, −8.310883614820891, −7.495150901424565, −6.894201468836049, −6.643523613670669, −5.412744784496184, −5.041147621297223, −4.750816566379049, −3.709293482943218, −2.860392852635747, −2.480984361913173, −1.516819562282572, −0.4974769642462355,
0.4974769642462355, 1.516819562282572, 2.480984361913173, 2.860392852635747, 3.709293482943218, 4.750816566379049, 5.041147621297223, 5.412744784496184, 6.643523613670669, 6.894201468836049, 7.495150901424565, 8.310883614820891, 8.594795071817180, 9.319356567538550, 10.05165247327406, 10.35015401203907, 11.09442277580184, 11.57431683155205, 12.00591680967286, 12.83171951157290, 13.25117331328908, 13.64734043138642, 14.24367038245539, 15.06843626882589, 15.23498116621228
