| L(s) = 1 | − 0.631·3-s − 3.16·7-s − 2.60·9-s + 1.90·11-s + 6.76·13-s − 0.737·17-s − 3.90·19-s + 1.99·21-s − 23-s + 3.53·27-s − 0.935·29-s − 1.02·31-s − 1.20·33-s − 5.94·37-s − 4.27·39-s + 11.0·41-s − 3.90·43-s + 5.47·47-s + 3.03·49-s + 0.465·51-s − 3.07·53-s + 2.46·57-s − 4.90·59-s + 11.0·61-s + 8.24·63-s − 3.20·67-s + 0.631·69-s + ⋯ |
| L(s) = 1 | − 0.364·3-s − 1.19·7-s − 0.867·9-s + 0.574·11-s + 1.87·13-s − 0.178·17-s − 0.895·19-s + 0.436·21-s − 0.208·23-s + 0.680·27-s − 0.173·29-s − 0.184·31-s − 0.209·33-s − 0.976·37-s − 0.684·39-s + 1.71·41-s − 0.595·43-s + 0.798·47-s + 0.433·49-s + 0.0651·51-s − 0.422·53-s + 0.326·57-s − 0.638·59-s + 1.41·61-s + 1.03·63-s − 0.391·67-s + 0.0760·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 0.631T + 3T^{2} \) |
| 7 | \( 1 + 3.16T + 7T^{2} \) |
| 11 | \( 1 - 1.90T + 11T^{2} \) |
| 13 | \( 1 - 6.76T + 13T^{2} \) |
| 17 | \( 1 + 0.737T + 17T^{2} \) |
| 19 | \( 1 + 3.90T + 19T^{2} \) |
| 29 | \( 1 + 0.935T + 29T^{2} \) |
| 31 | \( 1 + 1.02T + 31T^{2} \) |
| 37 | \( 1 + 5.94T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 3.90T + 43T^{2} \) |
| 47 | \( 1 - 5.47T + 47T^{2} \) |
| 53 | \( 1 + 3.07T + 53T^{2} \) |
| 59 | \( 1 + 4.90T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 + 3.20T + 67T^{2} \) |
| 71 | \( 1 + 8.86T + 71T^{2} \) |
| 73 | \( 1 - 9.40T + 73T^{2} \) |
| 79 | \( 1 + 1.56T + 79T^{2} \) |
| 83 | \( 1 + 2.64T + 83T^{2} \) |
| 89 | \( 1 + 18.6T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−7.19129645928260866732251439269, −6.47269429103235403811260766061, −6.07723354940678862060598793007, −5.63566386880028614419109092184, −4.48602814062654668803295152334, −3.71163170625325123692764237984, −3.23475953009717058303983811039, −2.22349093226576460215063467765, −1.07060145956354884422617677565, 0,
1.07060145956354884422617677565, 2.22349093226576460215063467765, 3.23475953009717058303983811039, 3.71163170625325123692764237984, 4.48602814062654668803295152334, 5.63566386880028614419109092184, 6.07723354940678862060598793007, 6.47269429103235403811260766061, 7.19129645928260866732251439269
