| L(s) = 1 | + 2.85·3-s + 3.60·5-s − 7-s + 5.17·9-s + 11-s + 13-s + 10.3·15-s + 2.10·17-s − 3.22·19-s − 2.85·21-s − 1.93·23-s + 7.99·25-s + 6.21·27-s + 0.252·29-s + 1.79·31-s + 2.85·33-s − 3.60·35-s − 1.21·37-s + 2.85·39-s + 10.4·41-s − 7.61·43-s + 18.6·45-s − 1.93·47-s + 49-s + 6.02·51-s + 6.07·53-s + 3.60·55-s + ⋯ |
| L(s) = 1 | + 1.65·3-s + 1.61·5-s − 0.377·7-s + 1.72·9-s + 0.301·11-s + 0.277·13-s + 2.66·15-s + 0.510·17-s − 0.739·19-s − 0.623·21-s − 0.403·23-s + 1.59·25-s + 1.19·27-s + 0.0469·29-s + 0.322·31-s + 0.497·33-s − 0.609·35-s − 0.199·37-s + 0.457·39-s + 1.62·41-s − 1.16·43-s + 2.77·45-s − 0.282·47-s + 0.142·49-s + 0.843·51-s + 0.834·53-s + 0.485·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.894802902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.894802902\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 2.85T + 3T^{2} \) |
| 5 | \( 1 - 3.60T + 5T^{2} \) |
| 17 | \( 1 - 2.10T + 17T^{2} \) |
| 19 | \( 1 + 3.22T + 19T^{2} \) |
| 23 | \( 1 + 1.93T + 23T^{2} \) |
| 29 | \( 1 - 0.252T + 29T^{2} \) |
| 31 | \( 1 - 1.79T + 31T^{2} \) |
| 37 | \( 1 + 1.21T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 7.61T + 43T^{2} \) |
| 47 | \( 1 + 1.93T + 47T^{2} \) |
| 53 | \( 1 - 6.07T + 53T^{2} \) |
| 59 | \( 1 + 0.0467T + 59T^{2} \) |
| 61 | \( 1 + 6.26T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 + 2.73T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 - 5.14T + 79T^{2} \) |
| 83 | \( 1 + 9.57T + 83T^{2} \) |
| 89 | \( 1 - 4.29T + 89T^{2} \) |
| 97 | \( 1 - 1.61T + 97T^{2} \) |
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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
−8.655950339629879232132875438085, −7.889243867924404879892897892596, −7.01072716115521793384268800238, −6.27952713565886456074490668497, −5.62337282992052013871188412198, −4.50420856634164696558456453007, −3.62349611103067920229890710647, −2.78839274458673937616925098927, −2.14281882369186131036977112635, −1.33966880874324325288976718365,
1.33966880874324325288976718365, 2.14281882369186131036977112635, 2.78839274458673937616925098927, 3.62349611103067920229890710647, 4.50420856634164696558456453007, 5.62337282992052013871188412198, 6.27952713565886456074490668497, 7.01072716115521793384268800238, 7.889243867924404879892897892596, 8.655950339629879232132875438085
