L(s) = 1 | + (0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + (−2.59 + 1.5i)5-s − 0.999i·6-s − 0.999i·8-s + (1 + 1.73i)9-s + (−1.5 + 2.59i)10-s + (−0.499 − 0.866i)12-s + (2 + 3i)13-s + 3i·15-s + (−0.5 − 0.866i)16-s + (−1.5 + 2.59i)17-s + (1.73 + i)18-s + (−5.19 + 3i)19-s + 3i·20-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + (−1.16 + 0.670i)5-s − 0.408i·6-s − 0.353i·8-s + (0.333 + 0.577i)9-s + (−0.474 + 0.821i)10-s + (−0.144 − 0.249i)12-s + (0.554 + 0.832i)13-s + 0.774i·15-s + (−0.125 − 0.216i)16-s + (−0.363 + 0.630i)17-s + (0.408 + 0.235i)18-s + (−1.19 + 0.688i)19-s + 0.670i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.842376394\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.842376394\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (-2 - 3i)T \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (2.59 - 1.5i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (5.19 - 3i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.59 + 1.5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + (-2.59 + 1.5i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.19 - 3i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.3 - 6i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 15iT - 71T^{2} \) |
| 73 | \( 1 + (5.19 + 3i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5 + 8.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + (5.19 - 3i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−10.01969343304365877293465048984, −8.743892942316142297163216514693, −8.074308903539308461126024310730, −7.16307316311745899471577015460, −6.67131606875721369278716354426, −5.54635315451879874415075692327, −4.21871839186642878911318151267, −3.83247627400737201861532665529, −2.60463291020006697312755133281, −1.55768486176936556977164529066,
0.62942755706693563744411160557, 2.71957943906692349476946821472, 3.72461181151864360026496024542, 4.39938797873491875689647558052, 5.03342860112668370402569229247, 6.32321271519708037450020845947, 7.04825075824595464390231614066, 8.096951569517084669789047790925, 8.607862651525854752172377921828, 9.344801839980331852773436266726