| L(s) = 1 | + 4·7-s − 4·11-s − 6·13-s + 17-s + 4·19-s − 4·23-s + 6·29-s − 4·31-s − 10·37-s + 6·41-s − 4·43-s − 8·47-s + 9·49-s + 6·53-s + 4·59-s − 14·61-s + 12·67-s + 12·71-s − 10·73-s − 16·77-s − 4·79-s + 4·83-s + 6·89-s − 24·91-s + 6·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1.20·11-s − 1.66·13-s + 0.242·17-s + 0.917·19-s − 0.834·23-s + 1.11·29-s − 0.718·31-s − 1.64·37-s + 0.937·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s + 0.824·53-s + 0.520·59-s − 1.79·61-s + 1.46·67-s + 1.42·71-s − 1.17·73-s − 1.82·77-s − 0.450·79-s + 0.439·83-s + 0.635·89-s − 2.51·91-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.833512210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.833512210\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.03127270583667, −14.48514410611996, −14.18314778863121, −13.69736355135794, −12.93674753155259, −12.34967375628108, −11.93070881405482, −11.46942510885297, −10.80482260018130, −10.20934746141851, −9.941395659846572, −9.136907313226689, −8.419617294759050, −7.962211037952860, −7.492198746900715, −7.118005898169455, −6.145965584919402, −5.260019727848312, −5.092020162519781, −4.623963015411183, −3.681113193444357, −2.848837385092241, −2.205775174061303, −1.613689685903388, −0.5036964002463846,
0.5036964002463846, 1.613689685903388, 2.205775174061303, 2.848837385092241, 3.681113193444357, 4.623963015411183, 5.092020162519781, 5.260019727848312, 6.145965584919402, 7.118005898169455, 7.492198746900715, 7.962211037952860, 8.419617294759050, 9.136907313226689, 9.941395659846572, 10.20934746141851, 10.80482260018130, 11.46942510885297, 11.93070881405482, 12.34967375628108, 12.93674753155259, 13.69736355135794, 14.18314778863121, 14.48514410611996, 15.03127270583667
