| L(s) = 1 | + 2.91·3-s + 2.91·7-s + 5.48·9-s + 0.598·11-s − 2.56·13-s + 2.91·17-s − 19-s + 8.48·21-s − 1.16·23-s + 7.22·27-s − 1.40·29-s + 7.88·31-s + 1.74·33-s − 6.53·37-s − 7.48·39-s + 9.48·41-s − 0.824·43-s + 6.05·47-s + 1.48·49-s + 8.48·51-s − 0.824·53-s − 2.91·57-s − 14.1·59-s − 1.88·61-s + 15.9·63-s + 5.11·67-s − 3.40·69-s + ⋯ |
| L(s) = 1 | + 1.68·3-s + 1.10·7-s + 1.82·9-s + 0.180·11-s − 0.712·13-s + 0.706·17-s − 0.229·19-s + 1.85·21-s − 0.243·23-s + 1.39·27-s − 0.260·29-s + 1.41·31-s + 0.303·33-s − 1.07·37-s − 1.19·39-s + 1.48·41-s − 0.125·43-s + 0.883·47-s + 0.211·49-s + 1.18·51-s − 0.113·53-s − 0.385·57-s − 1.84·59-s − 0.240·61-s + 2.01·63-s + 0.624·67-s − 0.409·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.662144259\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.662144259\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.91T + 3T^{2} \) |
| 7 | \( 1 - 2.91T + 7T^{2} \) |
| 11 | \( 1 - 0.598T + 11T^{2} \) |
| 13 | \( 1 + 2.56T + 13T^{2} \) |
| 17 | \( 1 - 2.91T + 17T^{2} \) |
| 23 | \( 1 + 1.16T + 23T^{2} \) |
| 29 | \( 1 + 1.40T + 29T^{2} \) |
| 31 | \( 1 - 7.88T + 31T^{2} \) |
| 37 | \( 1 + 6.53T + 37T^{2} \) |
| 41 | \( 1 - 9.48T + 41T^{2} \) |
| 43 | \( 1 + 0.824T + 43T^{2} \) |
| 47 | \( 1 - 6.05T + 47T^{2} \) |
| 53 | \( 1 + 0.824T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 + 1.88T + 61T^{2} \) |
| 67 | \( 1 - 5.11T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 - 1.03T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 7.70T + 97T^{2} \) |
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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
−8.997136773957987073767792355451, −8.481972547614559359798996326538, −7.66332254029826154735186500848, −7.37840270108168496588298059051, −6.06926343718301189502592623341, −4.86441556924652883129086445900, −4.19941150245915004172982549530, −3.16893294686290915309296093749, −2.32866663840395160526288463377, −1.40318524313527581858812944527,
1.40318524313527581858812944527, 2.32866663840395160526288463377, 3.16893294686290915309296093749, 4.19941150245915004172982549530, 4.86441556924652883129086445900, 6.06926343718301189502592623341, 7.37840270108168496588298059051, 7.66332254029826154735186500848, 8.481972547614559359798996326538, 8.997136773957987073767792355451
