| L(s) = 1 | − 1.59·2-s + 2.74·3-s + 0.529·4-s + 5-s − 4.36·6-s + 0.195·7-s + 2.33·8-s + 4.51·9-s − 1.59·10-s − 3.71·11-s + 1.45·12-s + 4.19·13-s − 0.311·14-s + 2.74·15-s − 4.77·16-s − 17-s − 7.18·18-s − 5.34·19-s + 0.529·20-s + 0.536·21-s + 5.90·22-s − 2.28·23-s + 6.40·24-s + 25-s − 6.66·26-s + 4.15·27-s + 0.103·28-s + ⋯ |
| L(s) = 1 | − 1.12·2-s + 1.58·3-s + 0.264·4-s + 0.447·5-s − 1.78·6-s + 0.0739·7-s + 0.826·8-s + 1.50·9-s − 0.502·10-s − 1.11·11-s + 0.419·12-s + 1.16·13-s − 0.0831·14-s + 0.707·15-s − 1.19·16-s − 0.242·17-s − 1.69·18-s − 1.22·19-s + 0.118·20-s + 0.116·21-s + 1.25·22-s − 0.475·23-s + 1.30·24-s + 0.200·25-s − 1.30·26-s + 0.799·27-s + 0.0195·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| good | 2 | \( 1 + 1.59T + 2T^{2} \) |
| 3 | \( 1 - 2.74T + 3T^{2} \) |
| 7 | \( 1 - 0.195T + 7T^{2} \) |
| 11 | \( 1 + 3.71T + 11T^{2} \) |
| 13 | \( 1 - 4.19T + 13T^{2} \) |
| 19 | \( 1 + 5.34T + 19T^{2} \) |
| 23 | \( 1 + 2.28T + 23T^{2} \) |
| 29 | \( 1 + 5.68T + 29T^{2} \) |
| 31 | \( 1 + 1.53T + 31T^{2} \) |
| 37 | \( 1 + 8.63T + 37T^{2} \) |
| 41 | \( 1 - 0.700T + 41T^{2} \) |
| 43 | \( 1 - 1.14T + 43T^{2} \) |
| 47 | \( 1 + 3.21T + 47T^{2} \) |
| 53 | \( 1 + 7.50T + 53T^{2} \) |
| 59 | \( 1 - 8.95T + 59T^{2} \) |
| 61 | \( 1 + 7.11T + 61T^{2} \) |
| 67 | \( 1 + 6.27T + 67T^{2} \) |
| 73 | \( 1 + 5.40T + 73T^{2} \) |
| 79 | \( 1 + 2.48T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 3.94T + 97T^{2} \) |
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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
−7.981355301890258354683441000257, −7.45820200308359665239233821641, −6.62086184000069345047689881632, −5.66546035919384981573094614610, −4.63810479785629173730238716630, −3.85988966488153683515669928191, −3.02165164718433638993446749393, −2.03010567267706948527573307502, −1.61464730764376440129137240421, 0,
1.61464730764376440129137240421, 2.03010567267706948527573307502, 3.02165164718433638993446749393, 3.85988966488153683515669928191, 4.63810479785629173730238716630, 5.66546035919384981573094614610, 6.62086184000069345047689881632, 7.45820200308359665239233821641, 7.981355301890258354683441000257
