| L(s) = 1 | + 1.61·3-s − 2.61·7-s − 0.381·9-s − 11-s + 3.47·13-s − 6.09·17-s + 8.23·19-s − 4.23·21-s + 2.61·23-s − 5.47·27-s − 7.32·29-s − 4.70·31-s − 1.61·33-s + 4.23·37-s + 5.61·39-s + 2.70·41-s − 10.9·43-s + 2.23·47-s − 0.145·49-s − 9.85·51-s − 1.38·53-s + 13.3·57-s − 2.70·59-s − 13.0·61-s + 0.999·63-s − 12·67-s + 4.23·69-s + ⋯ |
| L(s) = 1 | + 0.934·3-s − 0.989·7-s − 0.127·9-s − 0.301·11-s + 0.962·13-s − 1.47·17-s + 1.88·19-s − 0.924·21-s + 0.545·23-s − 1.05·27-s − 1.36·29-s − 0.845·31-s − 0.281·33-s + 0.696·37-s + 0.899·39-s + 0.422·41-s − 1.66·43-s + 0.326·47-s − 0.0208·49-s − 1.37·51-s − 0.189·53-s + 1.76·57-s − 0.352·59-s − 1.67·61-s + 0.125·63-s − 1.46·67-s + 0.509·69-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 7 | \( 1 + 2.61T + 7T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 + 6.09T + 17T^{2} \) |
| 19 | \( 1 - 8.23T + 19T^{2} \) |
| 23 | \( 1 - 2.61T + 23T^{2} \) |
| 29 | \( 1 + 7.32T + 29T^{2} \) |
| 31 | \( 1 + 4.70T + 31T^{2} \) |
| 37 | \( 1 - 4.23T + 37T^{2} \) |
| 41 | \( 1 - 2.70T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 2.23T + 47T^{2} \) |
| 53 | \( 1 + 1.38T + 53T^{2} \) |
| 59 | \( 1 + 2.70T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 9.38T + 73T^{2} \) |
| 79 | \( 1 - 9.61T + 79T^{2} \) |
| 83 | \( 1 - 6.56T + 83T^{2} \) |
| 89 | \( 1 - 1.32T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−7.981397268719135114321980059249, −7.41136882664921997670729273792, −6.58102285288908551089969903981, −5.86287780025624788595016204493, −5.05481401155434503390090986428, −3.88094413983543699643369094983, −3.30118260483432934557844944688, −2.66705662152233768551953581064, −1.55753257948447513813340114075, 0,
1.55753257948447513813340114075, 2.66705662152233768551953581064, 3.30118260483432934557844944688, 3.88094413983543699643369094983, 5.05481401155434503390090986428, 5.86287780025624788595016204493, 6.58102285288908551089969903981, 7.41136882664921997670729273792, 7.981397268719135114321980059249
