| L(s) = 1 | − 1.79·2-s + 1.20·4-s − 3·5-s + 7-s + 1.41·8-s + 5.37·10-s + 11-s + 13-s − 1.79·14-s − 4.95·16-s − 7.58·17-s − 6.58·19-s − 3.62·20-s − 1.79·22-s + 5.58·23-s + 4·25-s − 1.79·26-s + 1.20·28-s + 8.16·29-s + 3.58·31-s + 6.04·32-s + 13.5·34-s − 3·35-s + 37-s + 11.7·38-s − 4.25·40-s + 11.1·41-s + ⋯ |
| L(s) = 1 | − 1.26·2-s + 0.604·4-s − 1.34·5-s + 0.377·7-s + 0.501·8-s + 1.69·10-s + 0.301·11-s + 0.277·13-s − 0.478·14-s − 1.23·16-s − 1.83·17-s − 1.51·19-s − 0.810·20-s − 0.381·22-s + 1.16·23-s + 0.800·25-s − 0.351·26-s + 0.228·28-s + 1.51·29-s + 0.643·31-s + 1.06·32-s + 2.32·34-s − 0.507·35-s + 0.164·37-s + 1.91·38-s − 0.672·40-s + 1.74·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5122955597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5122955597\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 1.79T + 2T^{2} \) |
| 5 | \( 1 + 3T + 5T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 7.58T + 17T^{2} \) |
| 19 | \( 1 + 6.58T + 19T^{2} \) |
| 23 | \( 1 - 5.58T + 23T^{2} \) |
| 29 | \( 1 - 8.16T + 29T^{2} \) |
| 31 | \( 1 - 3.58T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 - 1.58T + 43T^{2} \) |
| 47 | \( 1 + 1.41T + 47T^{2} \) |
| 53 | \( 1 - 9.58T + 53T^{2} \) |
| 59 | \( 1 + 4.58T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 8.58T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 - 7.16T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 + 9.16T + 89T^{2} \) |
| 97 | \( 1 + 2.41T + 97T^{2} \) |
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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
−10.66913523976373548693074771689, −9.350105952498327959702765301853, −8.540067563817448895495835929655, −8.254905317298261597097947285807, −7.16575431738473006031599528774, −6.51170113575159198997059072050, −4.62752241095814088830050221313, −4.13592854823371659674728409848, −2.39181914087941912708702112316, −0.72687725945129004235660609447,
0.72687725945129004235660609447, 2.39181914087941912708702112316, 4.13592854823371659674728409848, 4.62752241095814088830050221313, 6.51170113575159198997059072050, 7.16575431738473006031599528774, 8.254905317298261597097947285807, 8.540067563817448895495835929655, 9.350105952498327959702765301853, 10.66913523976373548693074771689
