| L(s) = 1 | + (−1.12 − 0.794i)2-s + (−1.73 − 0.0319i)3-s + (−0.0321 − 0.0899i)4-s + (−3.71 + 2.05i)5-s + (1.92 + 1.41i)6-s + (2.64 − 1.34i)7-s + (−0.773 + 2.78i)8-s + (2.99 + 0.110i)9-s + (5.82 + 0.633i)10-s + (−0.838 + 0.316i)11-s + (0.0527 + 0.156i)12-s + (0.441 + 3.47i)13-s + (−4.04 − 0.589i)14-s + (6.50 − 3.44i)15-s + (2.93 − 2.40i)16-s + (−0.0255 − 0.472i)17-s + ⋯ |
| L(s) = 1 | + (−0.797 − 0.562i)2-s + (−0.999 − 0.0184i)3-s + (−0.0160 − 0.0449i)4-s + (−1.66 + 0.920i)5-s + (0.787 + 0.576i)6-s + (0.999 − 0.507i)7-s + (−0.273 + 0.985i)8-s + (0.999 + 0.0368i)9-s + (1.84 + 0.200i)10-s + (−0.252 + 0.0954i)11-s + (0.0152 + 0.0452i)12-s + (0.122 + 0.963i)13-s + (−1.08 − 0.157i)14-s + (1.67 − 0.889i)15-s + (0.734 − 0.601i)16-s + (−0.00620 − 0.114i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.173249 - 0.247864i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.173249 - 0.247864i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.73 + 0.0319i)T \) |
| 59 | \( 1 + (1.76 + 7.47i)T \) |
| good | 2 | \( 1 + (1.12 + 0.794i)T + (0.672 + 1.88i)T^{2} \) |
| 5 | \( 1 + (3.71 - 2.05i)T + (2.65 - 4.23i)T^{2} \) |
| 7 | \( 1 + (-2.64 + 1.34i)T + (4.13 - 5.64i)T^{2} \) |
| 11 | \( 1 + (0.838 - 0.316i)T + (8.25 - 7.27i)T^{2} \) |
| 13 | \( 1 + (-0.441 - 3.47i)T + (-12.5 + 3.25i)T^{2} \) |
| 17 | \( 1 + (0.0255 + 0.472i)T + (-16.9 + 1.83i)T^{2} \) |
| 19 | \( 1 + (4.35 - 4.12i)T + (1.02 - 18.9i)T^{2} \) |
| 23 | \( 1 + (-0.929 + 2.93i)T + (-18.8 - 13.2i)T^{2} \) |
| 29 | \( 1 + (0.742 + 8.19i)T + (-28.5 + 5.20i)T^{2} \) |
| 31 | \( 1 + (2.26 - 0.671i)T + (25.9 - 16.9i)T^{2} \) |
| 37 | \( 1 + (2.63 + 9.49i)T + (-31.7 + 19.0i)T^{2} \) |
| 41 | \( 1 + (0.652 - 0.713i)T + (-3.69 - 40.8i)T^{2} \) |
| 43 | \( 1 + (1.71 - 1.40i)T + (8.48 - 42.1i)T^{2} \) |
| 47 | \( 1 + (-4.19 - 2.32i)T + (24.9 + 39.8i)T^{2} \) |
| 53 | \( 1 + (-10.3 + 1.12i)T + (51.7 - 11.3i)T^{2} \) |
| 61 | \( 1 + (-4.52 - 3.18i)T + (20.5 + 57.4i)T^{2} \) |
| 67 | \( 1 + (-5.39 - 5.49i)T + (-1.20 + 66.9i)T^{2} \) |
| 71 | \( 1 + (-0.943 + 0.567i)T + (33.2 - 62.7i)T^{2} \) |
| 73 | \( 1 + (5.11 + 12.8i)T + (-52.9 + 50.2i)T^{2} \) |
| 79 | \( 1 + (9.59 + 8.45i)T + (9.95 + 78.3i)T^{2} \) |
| 83 | \( 1 + (-0.988 - 0.0714i)T + (82.1 + 11.9i)T^{2} \) |
| 89 | \( 1 + (7.32 + 3.38i)T + (57.6 + 67.8i)T^{2} \) |
| 97 | \( 1 + (-13.3 + 1.93i)T + (92.9 - 27.6i)T^{2} \) |
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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
−10.64027336858493107702064629916, −10.21768106408685461855688823039, −8.745491374276044638068898864131, −7.84870321072387116663230674118, −7.20043575494268697187254474329, −6.06601802441632307697817742782, −4.63073656645995566593983737283, −3.96189687684904176694329383495, −2.03195278777817985085665269809, −0.36032000988773258671321599935,
0.964356889583691721449403377993, 3.62981158226972908988409380014, 4.70256355187022529333170426216, 5.41581250469581585509542162671, 6.94490840565727639164973652484, 7.61369661743547310035270079925, 8.482749112164354424609361247722, 8.829446810241470846346562461453, 10.31900190583209894632473144147, 11.22945775143189131237054349677
