| L(s) = 1 | − 1.53·3-s − 1.74·5-s − 0.644·9-s + 1.47·11-s + 0.836·13-s + 2.68·15-s − 4.36·17-s + 0.873·19-s + 3.62·23-s − 1.93·25-s + 5.59·27-s + 5.63·29-s − 9.19·31-s − 2.25·33-s + 2.15·37-s − 1.28·39-s + 41-s − 1.50·43-s + 1.12·45-s − 8.53·47-s + 6.70·51-s + 7.12·53-s − 2.57·55-s − 1.34·57-s + 12.9·59-s + 12.0·61-s − 1.46·65-s + ⋯ |
| L(s) = 1 | − 0.886·3-s − 0.782·5-s − 0.214·9-s + 0.443·11-s + 0.231·13-s + 0.693·15-s − 1.05·17-s + 0.200·19-s + 0.754·23-s − 0.387·25-s + 1.07·27-s + 1.04·29-s − 1.65·31-s − 0.392·33-s + 0.354·37-s − 0.205·39-s + 0.156·41-s − 0.228·43-s + 0.168·45-s − 1.24·47-s + 0.938·51-s + 0.978·53-s − 0.347·55-s − 0.177·57-s + 1.68·59-s + 1.54·61-s − 0.181·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + 1.53T + 3T^{2} \) |
| 5 | \( 1 + 1.74T + 5T^{2} \) |
| 11 | \( 1 - 1.47T + 11T^{2} \) |
| 13 | \( 1 - 0.836T + 13T^{2} \) |
| 17 | \( 1 + 4.36T + 17T^{2} \) |
| 19 | \( 1 - 0.873T + 19T^{2} \) |
| 23 | \( 1 - 3.62T + 23T^{2} \) |
| 29 | \( 1 - 5.63T + 29T^{2} \) |
| 31 | \( 1 + 9.19T + 31T^{2} \) |
| 37 | \( 1 - 2.15T + 37T^{2} \) |
| 43 | \( 1 + 1.50T + 43T^{2} \) |
| 47 | \( 1 + 8.53T + 47T^{2} \) |
| 53 | \( 1 - 7.12T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 + 1.52T + 67T^{2} \) |
| 71 | \( 1 + 4.82T + 71T^{2} \) |
| 73 | \( 1 - 0.0120T + 73T^{2} \) |
| 79 | \( 1 - 2.55T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + 3.91T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−7.22034846552692352363630502313, −6.87573902256903566345916075311, −6.08182480558234827194012941285, −5.41930485807495609425869945622, −4.69995006301152166814596526382, −3.99328421702252865141051155035, −3.23852384205012912267991818634, −2.20181230899779175197731669170, −0.970610390182863561936795180274, 0,
0.970610390182863561936795180274, 2.20181230899779175197731669170, 3.23852384205012912267991818634, 3.99328421702252865141051155035, 4.69995006301152166814596526382, 5.41930485807495609425869945622, 6.08182480558234827194012941285, 6.87573902256903566345916075311, 7.22034846552692352363630502313
