L(s) = 1 | − 0.381·2-s − 3-s − 1.85·4-s + 2.23·5-s + 0.381·6-s + 1.47·8-s + 9-s − 0.854·10-s − 11-s + 1.85·12-s − 5.47·13-s − 2.23·15-s + 3.14·16-s + 6·17-s − 0.381·18-s + 0.236·19-s − 4.14·20-s + 0.381·22-s + 6.47·23-s − 1.47·24-s + 2.09·26-s − 27-s − 5.76·29-s + 0.854·30-s − 0.472·31-s − 4.14·32-s + 33-s + ⋯ |
L(s) = 1 | − 0.270·2-s − 0.577·3-s − 0.927·4-s + 0.999·5-s + 0.155·6-s + 0.520·8-s + 0.333·9-s − 0.270·10-s − 0.301·11-s + 0.535·12-s − 1.51·13-s − 0.577·15-s + 0.786·16-s + 1.45·17-s − 0.0900·18-s + 0.0541·19-s − 0.927·20-s + 0.0814·22-s + 1.34·23-s − 0.300·24-s + 0.409·26-s − 0.192·27-s − 1.07·29-s + 0.155·30-s − 0.0847·31-s − 0.732·32-s + 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.048843181\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.048843181\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 0.381T + 2T^{2} \) |
| 5 | \( 1 - 2.23T + 5T^{2} \) |
| 13 | \( 1 + 5.47T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 0.236T + 19T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 + 5.76T + 29T^{2} \) |
| 31 | \( 1 + 0.472T + 31T^{2} \) |
| 37 | \( 1 + 9.47T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 8.47T + 43T^{2} \) |
| 47 | \( 1 - 2.52T + 47T^{2} \) |
| 53 | \( 1 - 4.94T + 53T^{2} \) |
| 59 | \( 1 - 5.94T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 + 4.47T + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 - 6.47T + 79T^{2} \) |
| 83 | \( 1 - 3.52T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−9.443169202973369641617311594021, −8.903886452063052013559117840168, −7.62819729944523810633617193586, −7.22462958634975854523566779290, −5.81786533971778180846308007372, −5.36621248845182162621140308491, −4.68621960502611556592229805233, −3.43532907550187378214860793328, −2.11404604697061054501098014160, −0.78149473645515618841021118899,
0.78149473645515618841021118899, 2.11404604697061054501098014160, 3.43532907550187378214860793328, 4.68621960502611556592229805233, 5.36621248845182162621140308491, 5.81786533971778180846308007372, 7.22462958634975854523566779290, 7.62819729944523810633617193586, 8.903886452063052013559117840168, 9.443169202973369641617311594021