| L(s) = 1 | + (1.81 + 0.807i)2-s + (2.72 + 0.579i)3-s + (1.29 + 1.44i)4-s + (−0.00281 + 0.0267i)5-s + (4.47 + 3.25i)6-s + (−0.0378 − 0.116i)8-s + (4.35 + 1.93i)9-s + (−0.0267 + 0.0462i)10-s + (−1.92 − 2.70i)11-s + (2.70 + 4.67i)12-s + (−3.94 + 2.86i)13-s + (−0.0231 + 0.0713i)15-s + (0.430 − 4.09i)16-s + (−1.53 + 0.682i)17-s + (6.32 + 7.02i)18-s + (−0.919 + 1.02i)19-s + ⋯ |
| L(s) = 1 | + (1.28 + 0.570i)2-s + (1.57 + 0.334i)3-s + (0.648 + 0.720i)4-s + (−0.00125 + 0.0119i)5-s + (1.82 + 1.32i)6-s + (−0.0133 − 0.0411i)8-s + (1.45 + 0.645i)9-s + (−0.00844 + 0.0146i)10-s + (−0.580 − 0.814i)11-s + (0.779 + 1.35i)12-s + (−1.09 + 0.795i)13-s + (−0.00598 + 0.0184i)15-s + (0.107 − 1.02i)16-s + (−0.371 + 0.165i)17-s + (1.49 + 1.65i)18-s + (−0.210 + 0.234i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 - 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.648 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.70957 + 1.71391i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.70957 + 1.71391i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + (1.92 + 2.70i)T \) |
| good | 2 | \( 1 + (-1.81 - 0.807i)T + (1.33 + 1.48i)T^{2} \) |
| 3 | \( 1 + (-2.72 - 0.579i)T + (2.74 + 1.22i)T^{2} \) |
| 5 | \( 1 + (0.00281 - 0.0267i)T + (-4.89 - 1.03i)T^{2} \) |
| 13 | \( 1 + (3.94 - 2.86i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (1.53 - 0.682i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (0.919 - 1.02i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (4.03 + 6.98i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.97 - 6.08i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.419 - 3.98i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (0.509 - 0.108i)T + (33.8 - 15.0i)T^{2} \) |
| 41 | \( 1 + (-3.27 - 10.0i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 3.73T + 43T^{2} \) |
| 47 | \( 1 + (-5.99 + 6.65i)T + (-4.91 - 46.7i)T^{2} \) |
| 53 | \( 1 + (-0.413 - 3.93i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (6.51 + 7.23i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-0.885 + 8.42i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (1.40 - 2.43i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.65 + 1.20i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-7.00 - 7.78i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (-5.34 - 2.38i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-2.10 - 1.53i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.608 - 1.05i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.74 + 1.99i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.89812016715385245084014359464, −9.901148487715112537910046284230, −9.006021134828667258325567890927, −8.227547893899702381576011724917, −7.25502303789583334188200073015, −6.37066979135783191561142637951, −5.03088270096662350134286488853, −4.27707834375489971689441052748, −3.26349053282228359797447224789, −2.40323690814447976510083496331,
2.16026161222809437375457633486, 2.66256122584060957632950288615, 3.78404579856380677720239210349, 4.69283360008032691810715244660, 5.79649452467488801249423584775, 7.36318703486884647213867810665, 7.81877721422738697344991289118, 8.978109033383435213452428520094, 9.815370058840773344017931093178, 10.76890524970271734502388710332
