| L(s) = 1 | − 3·7-s − 4·13-s + 2·17-s + 2·19-s + 7·23-s + 8·29-s + 31-s + 3·37-s + 9·41-s + 9·43-s + 2·49-s + 53-s − 9·59-s + 14·61-s − 67-s − 4·71-s − 11·73-s + 16·79-s − 5·83-s + 12·91-s − 16·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 6·119-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 1.10·13-s + 0.485·17-s + 0.458·19-s + 1.45·23-s + 1.48·29-s + 0.179·31-s + 0.493·37-s + 1.40·41-s + 1.37·43-s + 2/7·49-s + 0.137·53-s − 1.17·59-s + 1.79·61-s − 0.122·67-s − 0.474·71-s − 1.28·73-s + 1.80·79-s − 0.548·83-s + 1.25·91-s − 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.550·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 241200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−13.04164093629517, −12.58254071864572, −12.30839002410177, −11.86369170120222, −11.26894992615701, −10.63533360484939, −10.44254383206765, −9.660818663530333, −9.443635013368667, −9.200609474716346, −8.343986640337818, −7.976080184374599, −7.326494960925737, −6.959766353800959, −6.577222494450656, −5.912500377810634, −5.489000082703536, −4.930284143909194, −4.360606654964438, −3.886717609178296, −2.992536484288770, −2.859783728183459, −2.371967036349629, −1.252077992194344, −0.8262496317855252, 0,
0.8262496317855252, 1.252077992194344, 2.371967036349629, 2.859783728183459, 2.992536484288770, 3.886717609178296, 4.360606654964438, 4.930284143909194, 5.489000082703536, 5.912500377810634, 6.577222494450656, 6.959766353800959, 7.326494960925737, 7.976080184374599, 8.343986640337818, 9.200609474716346, 9.443635013368667, 9.660818663530333, 10.44254383206765, 10.63533360484939, 11.26894992615701, 11.86369170120222, 12.30839002410177, 12.58254071864572, 13.04164093629517
