| L(s) = 1 | + 3.61·7-s + 11-s − 4.23·13-s − 2.85·17-s − 1.76·19-s + 4.70·23-s + 6.23·29-s − 10.0·31-s − 6.23·37-s − 2.09·41-s + 9.85·43-s − 2.23·47-s + 6.09·49-s + 3.38·53-s + 5.14·59-s − 2.85·61-s + 10.4·67-s + 9.56·71-s + 5.85·73-s + 3.61·77-s + 13.7·79-s + 12.0·83-s + 11·89-s − 15.3·91-s − 7.79·97-s + 16.9·101-s + 7.38·103-s + ⋯ |
| L(s) = 1 | + 1.36·7-s + 0.301·11-s − 1.17·13-s − 0.692·17-s − 0.404·19-s + 0.981·23-s + 1.15·29-s − 1.81·31-s − 1.02·37-s − 0.326·41-s + 1.50·43-s − 0.326·47-s + 0.870·49-s + 0.464·53-s + 0.669·59-s − 0.365·61-s + 1.27·67-s + 1.13·71-s + 0.685·73-s + 0.412·77-s + 1.54·79-s + 1.32·83-s + 1.16·89-s − 1.60·91-s − 0.791·97-s + 1.68·101-s + 0.727·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.253938509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.253938509\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 3.61T + 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + 4.23T + 13T^{2} \) |
| 17 | \( 1 + 2.85T + 17T^{2} \) |
| 19 | \( 1 + 1.76T + 19T^{2} \) |
| 23 | \( 1 - 4.70T + 23T^{2} \) |
| 29 | \( 1 - 6.23T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 6.23T + 37T^{2} \) |
| 41 | \( 1 + 2.09T + 41T^{2} \) |
| 43 | \( 1 - 9.85T + 43T^{2} \) |
| 47 | \( 1 + 2.23T + 47T^{2} \) |
| 53 | \( 1 - 3.38T + 53T^{2} \) |
| 59 | \( 1 - 5.14T + 59T^{2} \) |
| 61 | \( 1 + 2.85T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 9.56T + 71T^{2} \) |
| 73 | \( 1 - 5.85T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 - 11T + 89T^{2} \) |
| 97 | \( 1 + 7.79T + 97T^{2} \) |
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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
−7.77288916589702870102809899529, −7.04572919765603936892131118034, −6.56912221184652781840814351901, −5.38947060420747885596139063092, −5.02935715778046591858223521669, −4.36883671080791157387114132842, −3.53481967717745791530523888771, −2.39349678620666597099252761861, −1.88111596377499958923950916183, −0.72624096277977009652866925385,
0.72624096277977009652866925385, 1.88111596377499958923950916183, 2.39349678620666597099252761861, 3.53481967717745791530523888771, 4.36883671080791157387114132842, 5.02935715778046591858223521669, 5.38947060420747885596139063092, 6.56912221184652781840814351901, 7.04572919765603936892131118034, 7.77288916589702870102809899529
