| L(s) = 1 | + 0.618·3-s − 0.236·7-s − 2.61·9-s − 1.76·11-s − 4.61·13-s + 2.23·17-s − 7.09·19-s − 0.145·21-s + 0.763·23-s − 3.47·27-s − 5.76·29-s + 0.854·31-s − 1.09·33-s − 3.23·37-s − 2.85·39-s − 3·41-s + 10.7·43-s + 9.32·47-s − 6.94·49-s + 1.38·51-s − 3.14·53-s − 4.38·57-s − 5.61·59-s + 7.56·61-s + 0.618·63-s + 0.381·67-s + 0.472·69-s + ⋯ |
| L(s) = 1 | + 0.356·3-s − 0.0892·7-s − 0.872·9-s − 0.531·11-s − 1.28·13-s + 0.542·17-s − 1.62·19-s − 0.0318·21-s + 0.159·23-s − 0.668·27-s − 1.07·29-s + 0.153·31-s − 0.189·33-s − 0.532·37-s − 0.457·39-s − 0.468·41-s + 1.63·43-s + 1.36·47-s − 0.992·49-s + 0.193·51-s − 0.432·53-s − 0.580·57-s − 0.731·59-s + 0.968·61-s + 0.0778·63-s + 0.0466·67-s + 0.0568·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 - 0.618T + 3T^{2} \) |
| 7 | \( 1 + 0.236T + 7T^{2} \) |
| 11 | \( 1 + 1.76T + 11T^{2} \) |
| 13 | \( 1 + 4.61T + 13T^{2} \) |
| 17 | \( 1 - 2.23T + 17T^{2} \) |
| 19 | \( 1 + 7.09T + 19T^{2} \) |
| 23 | \( 1 - 0.763T + 23T^{2} \) |
| 29 | \( 1 + 5.76T + 29T^{2} \) |
| 31 | \( 1 - 0.854T + 31T^{2} \) |
| 37 | \( 1 + 3.23T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 9.32T + 47T^{2} \) |
| 53 | \( 1 + 3.14T + 53T^{2} \) |
| 59 | \( 1 + 5.61T + 59T^{2} \) |
| 61 | \( 1 - 7.56T + 61T^{2} \) |
| 67 | \( 1 - 0.381T + 67T^{2} \) |
| 71 | \( 1 + 4.23T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 + 6.23T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 + 8.47T + 89T^{2} \) |
| 97 | \( 1 - 8.56T + 97T^{2} \) |
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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
−9.482962187197362611598985387693, −8.742325814945750848456166769535, −7.913784338761383288647872299819, −7.18663091477197333862984853506, −6.04791213920734504841652067620, −5.24842649996125426865038520417, −4.18445501002436566871686361142, −2.97195030806430005927970865928, −2.13675778103966791857750734254, 0,
2.13675778103966791857750734254, 2.97195030806430005927970865928, 4.18445501002436566871686361142, 5.24842649996125426865038520417, 6.04791213920734504841652067620, 7.18663091477197333862984853506, 7.913784338761383288647872299819, 8.742325814945750848456166769535, 9.482962187197362611598985387693
