L(s) = 1 | + (1.24 + 1.04i)2-s + (−1.11 + 0.406i)3-s + (0.107 + 0.611i)4-s + (−1.80 − 0.658i)6-s + (1.11 + 1.93i)7-s + (1.11 − 1.93i)8-s + (−1.21 + 1.02i)9-s + (−2.82 + 4.88i)11-s + (−0.369 − 0.639i)12-s + (4.41 + 1.60i)13-s + (−0.627 + 3.55i)14-s + (4.56 − 1.66i)16-s + (0.601 + 0.505i)17-s − 2.56·18-s + (−2.63 + 3.47i)19-s + ⋯ |
L(s) = 1 | + (0.877 + 0.735i)2-s + (−0.644 + 0.234i)3-s + (0.0539 + 0.305i)4-s + (−0.738 − 0.268i)6-s + (0.421 + 0.730i)7-s + (0.394 − 0.683i)8-s + (−0.405 + 0.340i)9-s + (−0.850 + 1.47i)11-s + (−0.106 − 0.184i)12-s + (1.22 + 0.445i)13-s + (−0.167 + 0.951i)14-s + (1.14 − 0.415i)16-s + (0.146 + 0.122i)17-s − 0.605·18-s + (−0.603 + 0.797i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.402 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.936230 + 1.43493i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.936230 + 1.43493i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (2.63 - 3.47i)T \) |
good | 2 | \( 1 + (-1.24 - 1.04i)T + (0.347 + 1.96i)T^{2} \) |
| 3 | \( 1 + (1.11 - 0.406i)T + (2.29 - 1.92i)T^{2} \) |
| 7 | \( 1 + (-1.11 - 1.93i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.82 - 4.88i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.41 - 1.60i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.601 - 0.505i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.890 - 5.05i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.32 + 1.95i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (4.05 + 7.01i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 0.985T + 37T^{2} \) |
| 41 | \( 1 + (1.33 - 0.484i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.264 + 1.49i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (0.735 - 0.617i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (0.766 + 4.34i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-7.56 - 6.34i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.363 - 2.06i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.48 + 2.08i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.24 + 7.06i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-14.2 + 5.18i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (1.25 - 0.458i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (3.76 + 6.51i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.19 + 1.16i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-9.77 - 8.20i)T + (16.8 + 95.5i)T^{2} \) |
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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
−11.38423684095686788322120867962, −10.48785100349819829362721521899, −9.644623581103711445197235086505, −8.340020306992179327239220098941, −7.46438257830505592668517243400, −6.27653056161391134727193114432, −5.59261961204836585719194808507, −4.89653743006063284202802245021, −3.90602635785152016867137761994, −2.01130072139356479160717490692,
0.908776451496382630385220995431, 2.84156006611253946557338939168, 3.70792852528774303260998812250, 4.96335497901069045493254732790, 5.74208781234073788774106614870, 6.77128477631350869762131816930, 8.194079680433492323784792041691, 8.677248635781558647046649457784, 10.62604842686360597290906673495, 10.89347727115916676872548763265