| L(s) = 1 | − 0.488·2-s − 1.76·4-s − 1.92·5-s + 0.259·7-s + 1.83·8-s + 0.940·10-s − 11-s − 2.14·13-s − 0.126·14-s + 2.62·16-s − 4.30·17-s + 7.44·19-s + 3.39·20-s + 0.488·22-s − 3.82·23-s − 1.28·25-s + 1.04·26-s − 0.456·28-s + 3.58·29-s − 8.02·31-s − 4.95·32-s + 2.10·34-s − 0.499·35-s − 6.82·37-s − 3.63·38-s − 3.53·40-s − 5.88·41-s + ⋯ |
| L(s) = 1 | − 0.345·2-s − 0.880·4-s − 0.861·5-s + 0.0979·7-s + 0.649·8-s + 0.297·10-s − 0.301·11-s − 0.596·13-s − 0.0338·14-s + 0.656·16-s − 1.04·17-s + 1.70·19-s + 0.759·20-s + 0.104·22-s − 0.798·23-s − 0.257·25-s + 0.205·26-s − 0.0862·28-s + 0.665·29-s − 1.44·31-s − 0.875·32-s + 0.360·34-s − 0.0844·35-s − 1.12·37-s − 0.589·38-s − 0.559·40-s − 0.918·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5418453861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5418453861\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 0.488T + 2T^{2} \) |
| 5 | \( 1 + 1.92T + 5T^{2} \) |
| 7 | \( 1 - 0.259T + 7T^{2} \) |
| 13 | \( 1 + 2.14T + 13T^{2} \) |
| 17 | \( 1 + 4.30T + 17T^{2} \) |
| 19 | \( 1 - 7.44T + 19T^{2} \) |
| 23 | \( 1 + 3.82T + 23T^{2} \) |
| 29 | \( 1 - 3.58T + 29T^{2} \) |
| 31 | \( 1 + 8.02T + 31T^{2} \) |
| 37 | \( 1 + 6.82T + 37T^{2} \) |
| 41 | \( 1 + 5.88T + 41T^{2} \) |
| 43 | \( 1 - 7.55T + 43T^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 5.99T + 59T^{2} \) |
| 67 | \( 1 + 1.73T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 1.24T + 73T^{2} \) |
| 79 | \( 1 + 9.01T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 + 5.65T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−8.040077832191495611904159902507, −7.48287355318830021338404417605, −7.02228901510909089705617433504, −5.72042957658617347305882425777, −5.18688517201948765872237049806, −4.33528540099014361558318027172, −3.82371657221338043028051025113, −2.87446938556913760050719145681, −1.66876940068774431570543279170, −0.41203094428738351808325993555,
0.41203094428738351808325993555, 1.66876940068774431570543279170, 2.87446938556913760050719145681, 3.82371657221338043028051025113, 4.33528540099014361558318027172, 5.18688517201948765872237049806, 5.72042957658617347305882425777, 7.02228901510909089705617433504, 7.48287355318830021338404417605, 8.040077832191495611904159902507
