L(s) = 1 | + (−1.22 + 1.68i)2-s + (−2.09 + 0.679i)3-s + (−0.725 − 2.23i)4-s + (1.41 − 4.36i)6-s + 0.992i·7-s + (0.690 + 0.224i)8-s + (1.48 − 1.07i)9-s + (−1.61 − 1.17i)11-s + (3.03 + 4.17i)12-s + (−1.98 − 2.72i)13-s + (−1.67 − 1.21i)14-s + (2.57 − 1.87i)16-s + (2.75 + 0.894i)17-s + 3.82i·18-s + (0.798 − 2.45i)19-s + ⋯ |
L(s) = 1 | + (−0.866 + 1.19i)2-s + (−1.20 + 0.392i)3-s + (−0.362 − 1.11i)4-s + (0.578 − 1.78i)6-s + 0.375i·7-s + (0.244 + 0.0793i)8-s + (0.494 − 0.359i)9-s + (−0.487 − 0.354i)11-s + (0.876 + 1.20i)12-s + (−0.550 − 0.757i)13-s + (−0.447 − 0.325i)14-s + (0.643 − 0.467i)16-s + (0.667 + 0.216i)17-s + 0.901i·18-s + (0.183 − 0.563i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.375003 + 0.237984i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.375003 + 0.237984i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + (1.22 - 1.68i)T + (-0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (2.09 - 0.679i)T + (2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 - 0.992iT - 7T^{2} \) |
| 11 | \( 1 + (1.61 + 1.17i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (1.98 + 2.72i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-2.75 - 0.894i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.798 + 2.45i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-2.67 + 3.68i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.66 - 5.12i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.0421 + 0.129i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.26 - 1.73i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (6.98 - 5.07i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 4.64iT - 43T^{2} \) |
| 47 | \( 1 + (9.44 - 3.06i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-7.19 + 2.33i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.97 + 2.89i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.24 - 1.62i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.07 - 0.675i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-2.97 - 9.17i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.456 - 0.627i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (4.89 + 15.0i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-1.68 - 0.547i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-11.7 - 8.52i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-16.1 + 5.26i)T + (78.4 - 57.0i)T^{2} \) |
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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
−10.42122383148563814973300878599, −9.996293942626557413011493611835, −8.824985458984264269568485483248, −8.142247404129537557301986626619, −7.13406948165509570111648142415, −6.30996683335781526355380150729, −5.43396532419785136809919818624, −4.95670752359846846028010577918, −3.03614822265795697209152811876, −0.56598992628096162920116903435,
0.835232834462728783299950408826, 2.08697595013944376760068630204, 3.50407881407620518036839902784, 4.92790710026169120289135667723, 5.89934571640463029771246352479, 7.01380720808490597725850321110, 7.85866535370182641797401197881, 9.027566954687306607720612716520, 9.964683346968012825490144564222, 10.40555566955395208391517822763