| L(s) = 1 | − i·2-s + (−1.61 − 0.618i)3-s − 4-s + (−0.618 + 1.61i)6-s + (−2.61 + 0.381i)7-s + i·8-s + (2.23 + 2.00i)9-s + 4.47i·11-s + (1.61 + 0.618i)12-s − 1.23i·13-s + (0.381 + 2.61i)14-s + 16-s + 5.23·17-s + (2.00 − 2.23i)18-s − 8.47i·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (−0.934 − 0.356i)3-s − 0.5·4-s + (−0.252 + 0.660i)6-s + (−0.989 + 0.144i)7-s + 0.353i·8-s + (0.745 + 0.666i)9-s + 1.34i·11-s + (0.467 + 0.178i)12-s − 0.342i·13-s + (0.102 + 0.699i)14-s + 0.250·16-s + 1.26·17-s + (0.471 − 0.527i)18-s − 1.94i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8387162672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8387162672\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (1.61 + 0.618i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.61 - 0.381i)T \) |
| good | 11 | \( 1 - 4.47iT - 11T^{2} \) |
| 13 | \( 1 + 1.23iT - 13T^{2} \) |
| 17 | \( 1 - 5.23T + 17T^{2} \) |
| 19 | \( 1 + 8.47iT - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 7.70iT - 29T^{2} \) |
| 31 | \( 1 - 2.76iT - 31T^{2} \) |
| 37 | \( 1 + 0.763T + 37T^{2} \) |
| 41 | \( 1 - 2.47T + 41T^{2} \) |
| 43 | \( 1 - 4.94T + 43T^{2} \) |
| 47 | \( 1 + 6.47T + 47T^{2} \) |
| 53 | \( 1 + 0.472iT - 53T^{2} \) |
| 59 | \( 1 + 4.47T + 59T^{2} \) |
| 61 | \( 1 + 7.23iT - 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 7.23iT - 71T^{2} \) |
| 73 | \( 1 + 11.2iT - 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 5.52T + 89T^{2} \) |
| 97 | \( 1 - 0.763iT - 97T^{2} \) |
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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
−9.764042201512197154020295855658, −9.316775630781878120071989023339, −7.84228382843941592195317192622, −7.13330991179478640006811743084, −6.24166989447364325111266530492, −5.26672732927500359932293631356, −4.48802128430245909023111308315, −3.21877736379699181694207944397, −2.05948800320747712998092132007, −0.57542410400066385690911800806,
0.967212131700865204571020436913, 3.33038250647856729577735503743, 3.97914396270348012768379419976, 5.32513458168058395407693369992, 5.94551444455001453471782069814, 6.50770455237962211578120041866, 7.51596801192162309223388720226, 8.466041748524274438081425607461, 9.402304736577377023567043038608, 10.15388647975471956741303365188
