| L(s) = 1 | + (−0.366 − 0.930i)2-s + (0.0596 + 1.27i)3-s + (−0.731 + 0.681i)4-s + (−2.72 − 1.54i)5-s + (1.16 − 0.520i)6-s + (0.380 + 0.338i)7-s + (0.902 + 0.430i)8-s + (1.37 − 0.129i)9-s + (−0.438 + 3.09i)10-s + (−0.0463 + 0.393i)11-s + (−0.909 − 0.888i)12-s + (−1.38 − 0.195i)13-s + (0.175 − 0.478i)14-s + (1.79 − 3.55i)15-s + (0.0702 − 0.997i)16-s + (−2.80 + 0.330i)17-s + ⋯ |
| L(s) = 1 | + (−0.259 − 0.657i)2-s + (0.0344 + 0.733i)3-s + (−0.365 + 0.340i)4-s + (−1.21 − 0.690i)5-s + (0.473 − 0.212i)6-s + (0.143 + 0.127i)7-s + (0.319 + 0.152i)8-s + (0.459 − 0.0431i)9-s + (−0.138 + 0.980i)10-s + (−0.0139 + 0.118i)11-s + (−0.262 − 0.256i)12-s + (−0.383 − 0.0543i)13-s + (0.0469 − 0.127i)14-s + (0.464 − 0.916i)15-s + (0.0175 − 0.249i)16-s + (−0.680 + 0.0801i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.851 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0280617 + 0.0989237i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0280617 + 0.0989237i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.366 + 0.930i)T \) |
| 269 | \( 1 + (12.5 + 10.6i)T \) |
| good | 3 | \( 1 + (-0.0596 - 1.27i)T + (-2.98 + 0.280i)T^{2} \) |
| 5 | \( 1 + (2.72 + 1.54i)T + (2.56 + 4.29i)T^{2} \) |
| 7 | \( 1 + (-0.380 - 0.338i)T + (0.818 + 6.95i)T^{2} \) |
| 11 | \( 1 + (0.0463 - 0.393i)T + (-10.6 - 2.55i)T^{2} \) |
| 13 | \( 1 + (1.38 + 0.195i)T + (12.4 + 3.60i)T^{2} \) |
| 17 | \( 1 + (2.80 - 0.330i)T + (16.5 - 3.94i)T^{2} \) |
| 19 | \( 1 + (7.06 - 2.97i)T + (13.2 - 13.5i)T^{2} \) |
| 23 | \( 1 + (6.02 - 0.282i)T + (22.8 - 2.15i)T^{2} \) |
| 29 | \( 1 + (4.62 + 2.76i)T + (13.7 + 25.5i)T^{2} \) |
| 31 | \( 1 + (0.742 - 3.10i)T + (-27.6 - 14.0i)T^{2} \) |
| 37 | \( 1 + (1.72 + 2.60i)T + (-14.3 + 34.0i)T^{2} \) |
| 41 | \( 1 + (-0.255 - 0.100i)T + (29.9 + 27.9i)T^{2} \) |
| 43 | \( 1 + (3.23 - 5.39i)T + (-20.3 - 37.8i)T^{2} \) |
| 47 | \( 1 + (8.22 + 5.18i)T + (20.2 + 42.4i)T^{2} \) |
| 53 | \( 1 + (-2.97 + 9.44i)T + (-43.4 - 30.3i)T^{2} \) |
| 59 | \( 1 + (-3.00 - 3.22i)T + (-4.14 + 58.8i)T^{2} \) |
| 61 | \( 1 + (2.71 - 12.6i)T + (-55.6 - 24.9i)T^{2} \) |
| 67 | \( 1 + (-7.13 - 6.65i)T + (4.70 + 66.8i)T^{2} \) |
| 71 | \( 1 + (-13.5 + 4.95i)T + (54.1 - 45.9i)T^{2} \) |
| 73 | \( 1 + (-2.04 - 1.73i)T + (11.9 + 72.0i)T^{2} \) |
| 79 | \( 1 + (-0.230 - 0.874i)T + (-68.7 + 38.9i)T^{2} \) |
| 83 | \( 1 + (-1.71 - 9.05i)T + (-77.2 + 30.4i)T^{2} \) |
| 89 | \( 1 + (-9.36 + 3.18i)T + (70.5 - 54.3i)T^{2} \) |
| 97 | \( 1 + (-0.967 - 4.51i)T + (-88.4 + 39.7i)T^{2} \) |
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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
−11.14859501536255618715700562613, −10.30196083250650532195935335481, −9.562768202586686500517078230441, −8.508749275777667777461557877905, −8.052849252431596777938771465534, −6.81762864563877770489309866231, −5.18453053180280053420683360132, −4.17358615456059766069576869035, −3.81671785282680783118564468285, −1.97432210297087580205655613919,
0.06248520366894174854818861602, 2.12396710371024180424265753494, 3.82204046288545426873555505787, 4.70355726969184133413117974337, 6.34573046379579159668058989982, 6.89654218875604079140944226427, 7.73716526775436803436890717651, 8.249653879424758118486053998173, 9.427551707054128581989943169846, 10.59949483651409781080746782020
