| L(s) = 1 | + 7-s + 11-s − 6·13-s + 2·17-s + 2·19-s − 6·23-s + 4·37-s + 8·41-s + 4·43-s − 4·47-s + 49-s + 8·53-s + 4·59-s + 14·61-s − 8·67-s − 12·71-s − 2·73-s + 77-s + 6·79-s + 2·89-s − 6·91-s − 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.301·11-s − 1.66·13-s + 0.485·17-s + 0.458·19-s − 1.25·23-s + 0.657·37-s + 1.24·41-s + 0.609·43-s − 0.583·47-s + 1/7·49-s + 1.09·53-s + 0.520·59-s + 1.79·61-s − 0.977·67-s − 1.42·71-s − 0.234·73-s + 0.113·77-s + 0.675·79-s + 0.211·89-s − 0.628·91-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.097125612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.097125612\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−13.37575966959197, −12.94962405561419, −12.31779724669382, −11.98974813608898, −11.65623553819516, −11.08697026119751, −10.43099944796564, −9.986457076105491, −9.650536759279315, −9.145679976848346, −8.515579534708101, −7.962834517419960, −7.483687795156029, −7.224961537157762, −6.491153292568426, −5.882415474071352, −5.429270736420188, −4.909679203040725, −4.246359039324965, −3.932358943389950, −3.046234453772370, −2.488773394828269, −2.013967499965706, −1.181696414935612, −0.4543142706291910,
0.4543142706291910, 1.181696414935612, 2.013967499965706, 2.488773394828269, 3.046234453772370, 3.932358943389950, 4.246359039324965, 4.909679203040725, 5.429270736420188, 5.882415474071352, 6.491153292568426, 7.224961537157762, 7.483687795156029, 7.962834517419960, 8.515579534708101, 9.145679976848346, 9.650536759279315, 9.986457076105491, 10.43099944796564, 11.08697026119751, 11.65623553819516, 11.98974813608898, 12.31779724669382, 12.94962405561419, 13.37575966959197
