L(s) = 1 | + i·2-s + (1.68 + 0.395i)3-s − 4-s + (−0.866 − 0.5i)5-s + (−0.395 + 1.68i)6-s + (−1.28 − 2.22i)7-s − i·8-s + (2.68 + 1.33i)9-s + (0.5 − 0.866i)10-s + (−2.56 + 4.44i)11-s + (−1.68 − 0.395i)12-s + (4.59 + 2.65i)13-s + (2.22 − 1.28i)14-s + (−1.26 − 1.18i)15-s + 16-s + (2.59 + 4.49i)17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.973 + 0.228i)3-s − 0.5·4-s + (−0.387 − 0.223i)5-s + (−0.161 + 0.688i)6-s + (−0.486 − 0.842i)7-s − 0.353i·8-s + (0.895 + 0.444i)9-s + (0.158 − 0.273i)10-s + (−0.774 + 1.34i)11-s + (−0.486 − 0.114i)12-s + (1.27 + 0.735i)13-s + (0.595 − 0.343i)14-s + (−0.325 − 0.306i)15-s + 0.250·16-s + (0.629 + 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.19587 + 1.39980i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.19587 + 1.39980i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.68 - 0.395i)T \) |
| 5 | \( 1 + (0.866 + 0.5i)T \) |
| 31 | \( 1 + (-0.790 - 5.51i)T \) |
good | 7 | \( 1 + (1.28 + 2.22i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.56 - 4.44i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.59 - 2.65i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.59 - 4.49i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.264 + 0.458i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 3.03T + 23T^{2} \) |
| 29 | \( 1 - 8.44T + 29T^{2} \) |
| 37 | \( 1 + (-0.580 + 0.335i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-9.85 - 5.68i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6.69 - 3.86i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 6.75iT - 47T^{2} \) |
| 53 | \( 1 + (0.822 - 1.42i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (10.4 - 6.03i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 7.06iT - 61T^{2} \) |
| 67 | \( 1 + (-3.98 + 6.89i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (8.45 + 4.88i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.21 + 1.85i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.75 + 5.05i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.97 + 6.87i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 + 6.46T + 97T^{2} \) |
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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
−10.16402330591034365904168868004, −9.366956518617655039021636901095, −8.393750063952909490697233031916, −7.893381250855685689836307221613, −7.03953881922641736949326007191, −6.25114761020524842184946701126, −4.72713738255230226576157270416, −4.15182241593301157505986403664, −3.21443834991166630094257636450, −1.53984415476898107354531940710,
0.854468408849696172655494559852, 2.67553142441107073190025572330, 3.05173258605066781329148614541, 4.03077903583713429414610631567, 5.50878080160960629109949259463, 6.29878967979187111432593436692, 7.71285932397124012639914558711, 8.302597848538350907639012803328, 8.923943490506628206919698072619, 9.828837455068276875131314930721