L(s) = 1 | + (−1.36 + 0.989i)2-s + (−0.219 + 0.675i)3-s + (0.257 − 0.793i)4-s + (−0.369 − 1.13i)6-s − 4.59·7-s + (−0.606 − 1.86i)8-s + (2.01 + 1.46i)9-s + (−3.16 + 2.30i)11-s + (0.479 + 0.348i)12-s + (0.463 + 0.336i)13-s + (6.25 − 4.54i)14-s + (4.02 + 2.92i)16-s + (−0.0718 − 0.221i)17-s − 4.20·18-s + (−1.71 − 5.27i)19-s + ⋯ |
L(s) = 1 | + (−0.963 + 0.699i)2-s + (−0.126 + 0.390i)3-s + (0.128 − 0.396i)4-s + (−0.150 − 0.464i)6-s − 1.73·7-s + (−0.214 − 0.660i)8-s + (0.672 + 0.488i)9-s + (−0.954 + 0.693i)11-s + (0.138 + 0.100i)12-s + (0.128 + 0.0934i)13-s + (1.67 − 1.21i)14-s + (1.00 + 0.730i)16-s + (−0.0174 − 0.0536i)17-s − 0.990·18-s + (−0.393 − 1.21i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.242089 - 0.0900322i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.242089 - 0.0900322i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.36 - 0.989i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (0.219 - 0.675i)T + (-2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 + 4.59T + 7T^{2} \) |
| 11 | \( 1 + (3.16 - 2.30i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.463 - 0.336i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.0718 + 0.221i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (1.71 + 5.27i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-3.99 + 2.89i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.27 + 3.92i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.08 + 3.32i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-4.38 - 3.18i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (8.43 + 6.12i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 1.38T + 43T^{2} \) |
| 47 | \( 1 + (-0.284 + 0.875i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.380 + 1.17i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (3.64 + 2.64i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-9.43 + 6.85i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (0.912 + 2.80i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-0.990 + 3.04i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-8.32 + 6.04i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (2.97 - 9.14i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (3.23 + 9.97i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (5.87 - 4.26i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (2.57 - 7.91i)T + (-78.4 - 57.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.09846290479226432021831918552, −9.664420895810163400576321579246, −8.894971037600060344638773561005, −7.81914902757744140387929586170, −6.95866878671233471671837076801, −6.44853522756827634948765440386, −5.08239316000988705452575218475, −3.90473805485733071188866081183, −2.62592484420435067692710133639, −0.22215528853858164029356427333,
1.17494026836269908683895571392, 2.75526581529901661945563982762, 3.63333356442962856588125455233, 5.46460525331065998497114086209, 6.30361154683144594021222945892, 7.25945225525606926186512706181, 8.324792723013906478973447370637, 9.174986575139578816142122496473, 9.975083673753085432392305268247, 10.39136398580322419241962822197