| L(s) = 1 | + 2-s − 3.14·3-s + 4-s + 0.363·5-s − 3.14·6-s + 8-s + 6.86·9-s + 0.363·10-s − 1.50·11-s − 3.14·12-s − 1.14·15-s + 16-s + 5.14·17-s + 6.86·18-s + 4.28·19-s + 0.363·20-s − 1.50·22-s + 2.72·23-s − 3.14·24-s − 4.86·25-s − 12.1·27-s + 2.36·29-s − 1.14·30-s − 5.14·31-s + 32-s + 4.72·33-s + 5.14·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.81·3-s + 0.5·4-s + 0.162·5-s − 1.28·6-s + 0.353·8-s + 2.28·9-s + 0.114·10-s − 0.453·11-s − 0.906·12-s − 0.294·15-s + 0.250·16-s + 1.24·17-s + 1.61·18-s + 0.982·19-s + 0.0812·20-s − 0.320·22-s + 0.568·23-s − 0.641·24-s − 0.973·25-s − 2.33·27-s + 0.438·29-s − 0.208·30-s − 0.923·31-s + 0.176·32-s + 0.822·33-s + 0.881·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.651495173\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.651495173\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + 3.14T + 3T^{2} \) |
| 5 | \( 1 - 0.363T + 5T^{2} \) |
| 11 | \( 1 + 1.50T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 5.14T + 17T^{2} \) |
| 19 | \( 1 - 4.28T + 19T^{2} \) |
| 23 | \( 1 - 2.72T + 23T^{2} \) |
| 29 | \( 1 - 2.36T + 29T^{2} \) |
| 31 | \( 1 + 5.14T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 - 5.37T + 53T^{2} \) |
| 59 | \( 1 + 4.23T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 15.0T + 67T^{2} \) |
| 71 | \( 1 + 8.87T + 71T^{2} \) |
| 73 | \( 1 - 3.71T + 73T^{2} \) |
| 79 | \( 1 - 2.13T + 79T^{2} \) |
| 83 | \( 1 - 6.77T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 4.13T + 97T^{2} \) |
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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
−8.149234024983348468879278385663, −7.37752997701067641948445898943, −6.78878321813306502837326791991, −5.98890164507959418644104762611, −5.34231240505366626727734684273, −5.10553567378500718827325523978, −4.05629103304253146510315439106, −3.18401666713030731483646301308, −1.78198599507577835701458639944, −0.74804007563192242623043442198,
0.74804007563192242623043442198, 1.78198599507577835701458639944, 3.18401666713030731483646301308, 4.05629103304253146510315439106, 5.10553567378500718827325523978, 5.34231240505366626727734684273, 5.98890164507959418644104762611, 6.78878321813306502837326791991, 7.37752997701067641948445898943, 8.149234024983348468879278385663
