| L(s) = 1 | − 9.39·3-s + 25.2·7-s + 61.3·9-s + 25.8·11-s + 76.7·13-s − 6.74·17-s − 65.0·19-s − 237.·21-s − 141.·23-s − 322.·27-s − 19.6·29-s − 276.·31-s − 243.·33-s − 298.·37-s − 720.·39-s − 79.6·41-s + 472.·43-s − 46.2·47-s + 295.·49-s + 63.3·51-s + 682.·53-s + 611.·57-s − 95.8·59-s + 58.8·61-s + 1.54e3·63-s − 911.·67-s + 1.33e3·69-s + ⋯ |
| L(s) = 1 | − 1.80·3-s + 1.36·7-s + 2.27·9-s + 0.709·11-s + 1.63·13-s − 0.0961·17-s − 0.785·19-s − 2.46·21-s − 1.28·23-s − 2.29·27-s − 0.125·29-s − 1.60·31-s − 1.28·33-s − 1.32·37-s − 2.95·39-s − 0.303·41-s + 1.67·43-s − 0.143·47-s + 0.861·49-s + 0.173·51-s + 1.76·53-s + 1.42·57-s − 0.211·59-s + 0.123·61-s + 3.09·63-s − 1.66·67-s + 2.32·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 9.39T + 27T^{2} \) |
| 7 | \( 1 - 25.2T + 343T^{2} \) |
| 11 | \( 1 - 25.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 76.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 6.74T + 4.91e3T^{2} \) |
| 19 | \( 1 + 65.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 141.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 19.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 276.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 298.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 79.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 472.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 46.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 682.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 95.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 58.8T + 2.26e5T^{2} \) |
| 67 | \( 1 + 911.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.05e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 641.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 908.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 121.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 622.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.12e3T + 9.12e5T^{2} \) |
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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
−8.421546479915292172895008418996, −7.44704218120983459942262481076, −6.65735037485323668184105961961, −5.83326193644456546336822755904, −5.46250641808233501788468766772, −4.30629829961651196817188528914, −3.94355657105766456086623085324, −1.79803527388642142730134536638, −1.24338626758048972263158180693, 0,
1.24338626758048972263158180693, 1.79803527388642142730134536638, 3.94355657105766456086623085324, 4.30629829961651196817188528914, 5.46250641808233501788468766772, 5.83326193644456546336822755904, 6.65735037485323668184105961961, 7.44704218120983459942262481076, 8.421546479915292172895008418996
