L(s) = 1 | + 0.618·3-s + 0.381·7-s − 2.61·9-s − 11-s + 5.47·13-s − 5.09·17-s + 3.76·19-s + 0.236·21-s − 0.381·23-s − 3.47·27-s + 8.32·29-s + 8.70·31-s − 0.618·33-s + 0.236·37-s + 3.38·39-s − 10.7·41-s − 6.94·43-s + 2.23·47-s − 6.85·49-s − 3.14·51-s + 3.61·53-s + 2.32·57-s + 10.7·59-s − 1.90·61-s − 63-s + 12·67-s − 0.236·69-s + ⋯ |
L(s) = 1 | + 0.356·3-s + 0.144·7-s − 0.872·9-s − 0.301·11-s + 1.51·13-s − 1.23·17-s + 0.863·19-s + 0.0515·21-s − 0.0796·23-s − 0.668·27-s + 1.54·29-s + 1.56·31-s − 0.107·33-s + 0.0388·37-s + 0.541·39-s − 1.67·41-s − 1.05·43-s + 0.326·47-s − 0.979·49-s − 0.440·51-s + 0.496·53-s + 0.308·57-s + 1.39·59-s − 0.244·61-s − 0.125·63-s + 1.46·67-s − 0.0284·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.098542295\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.098542295\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - 0.618T + 3T^{2} \) |
| 7 | \( 1 - 0.381T + 7T^{2} \) |
| 13 | \( 1 - 5.47T + 13T^{2} \) |
| 17 | \( 1 + 5.09T + 17T^{2} \) |
| 19 | \( 1 - 3.76T + 19T^{2} \) |
| 23 | \( 1 + 0.381T + 23T^{2} \) |
| 29 | \( 1 - 8.32T + 29T^{2} \) |
| 31 | \( 1 - 8.70T + 31T^{2} \) |
| 37 | \( 1 - 0.236T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 + 6.94T + 43T^{2} \) |
| 47 | \( 1 - 2.23T + 47T^{2} \) |
| 53 | \( 1 - 3.61T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 1.90T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 8.18T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 7.38T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−8.549086188306114807515407536749, −7.897469879952028997574627456276, −6.68809024919949034091786092554, −6.35840694464300036870193010589, −5.36012444384650687012821119510, −4.67132293343261082011012416090, −3.63079319877428014649182622740, −2.97980383295172907167557654382, −2.03583104278383959820010443173, −0.802396854074297544795514332761,
0.802396854074297544795514332761, 2.03583104278383959820010443173, 2.97980383295172907167557654382, 3.63079319877428014649182622740, 4.67132293343261082011012416090, 5.36012444384650687012821119510, 6.35840694464300036870193010589, 6.68809024919949034091786092554, 7.897469879952028997574627456276, 8.549086188306114807515407536749