L(s) = 1 | + (−0.841 + 0.540i)3-s + (1.21 + 2.67i)5-s + (0.449 − 0.132i)7-s + (0.415 − 0.909i)9-s + (0.144 + 0.166i)11-s + (2.44 + 0.716i)13-s + (−2.46 − 1.58i)15-s + (−0.514 + 3.58i)17-s + (0.278 + 1.93i)19-s + (−0.306 + 0.354i)21-s + (−4.77 + 0.426i)23-s + (−2.36 + 2.73i)25-s + (0.142 + 0.989i)27-s + (−0.638 + 4.43i)29-s + (−3.58 − 2.30i)31-s + ⋯ |
L(s) = 1 | + (−0.485 + 0.312i)3-s + (0.545 + 1.19i)5-s + (0.169 − 0.0498i)7-s + (0.138 − 0.303i)9-s + (0.0434 + 0.0501i)11-s + (0.676 + 0.198i)13-s + (−0.637 − 0.409i)15-s + (−0.124 + 0.868i)17-s + (0.0639 + 0.444i)19-s + (−0.0669 + 0.0772i)21-s + (−0.996 + 0.0889i)23-s + (−0.473 + 0.546i)25-s + (0.0273 + 0.190i)27-s + (−0.118 + 0.824i)29-s + (−0.644 − 0.414i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0771 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0771 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.876930 + 0.947433i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.876930 + 0.947433i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.841 - 0.540i)T \) |
| 23 | \( 1 + (4.77 - 0.426i)T \) |
good | 5 | \( 1 + (-1.21 - 2.67i)T + (-3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-0.449 + 0.132i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-0.144 - 0.166i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.44 - 0.716i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.514 - 3.58i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.278 - 1.93i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (0.638 - 4.43i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (3.58 + 2.30i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-0.241 + 0.528i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-4.48 - 9.82i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-0.272 + 0.175i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 3.28T + 47T^{2} \) |
| 53 | \( 1 + (-3.68 + 1.08i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-8.26 - 2.42i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-0.814 - 0.523i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-4.41 + 5.09i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (4.30 - 4.97i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (1.56 + 10.8i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (5.30 + 1.55i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-3.14 + 6.89i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-11.7 + 7.57i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (3.91 + 8.56i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−10.89816210395738326748369399428, −10.30480674693065794813110832987, −9.510269366374136246634296564982, −8.357825215856730549698592170655, −7.26729142097126357426184351219, −6.27610350648427851759660914944, −5.77337949759781388803018728571, −4.31200630107732699365496819815, −3.27532545768416905374826867642, −1.81743495695407406884294034879,
0.825494586346199931197744751085, 2.18396823749680259217600835102, 4.01018437106311666030074433380, 5.14826493642596053047054514183, 5.74480429490672947033534486695, 6.85124804631276490883883676408, 7.967016949216427587261452660683, 8.836204055219185409288351111445, 9.572201179690667591209277132631, 10.60009875997299964491321502987