L(s) = 1 | + (12.0 + 12.0i)3-s + (−34.4 − 44.0i)5-s + (−74.8 + 74.8i)7-s + 49.4i·9-s + 432. i·11-s + (639. − 639. i)13-s + (116. − 948. i)15-s + (−1.42e3 − 1.42e3i)17-s − 2.37e3·19-s − 1.81e3·21-s + (170. + 170. i)23-s + (−757. + 3.03e3i)25-s + (2.34e3 − 2.34e3i)27-s − 5.00e3i·29-s − 7.16e3i·31-s + ⋯ |
L(s) = 1 | + (0.775 + 0.775i)3-s + (−0.615 − 0.788i)5-s + (−0.577 + 0.577i)7-s + 0.203i·9-s + 1.07i·11-s + (1.04 − 1.04i)13-s + (0.133 − 1.08i)15-s + (−1.19 − 1.19i)17-s − 1.50·19-s − 0.895·21-s + (0.0671 + 0.0671i)23-s + (−0.242 + 0.970i)25-s + (0.617 − 0.617i)27-s − 1.10i·29-s − 1.33i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.346 + 0.937i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.346 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.8704608064\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8704608064\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (34.4 + 44.0i)T \) |
good | 3 | \( 1 + (-12.0 - 12.0i)T + 243iT^{2} \) |
| 7 | \( 1 + (74.8 - 74.8i)T - 1.68e4iT^{2} \) |
| 11 | \( 1 - 432. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-639. + 639. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (1.42e3 + 1.42e3i)T + 1.41e6iT^{2} \) |
| 19 | \( 1 + 2.37e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-170. - 170. i)T + 6.43e6iT^{2} \) |
| 29 | \( 1 + 5.00e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 7.16e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (4.64e3 + 4.64e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 - 1.57e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + (4.52e3 + 4.52e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + (-489. + 489. i)T - 2.29e8iT^{2} \) |
| 53 | \( 1 + (1.53e3 - 1.53e3i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 - 2.54e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.14e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (3.79e4 - 3.79e4i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 + 4.14e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-3.76e4 + 3.76e4i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 - 3.97e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (4.03e4 + 4.03e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 + 4.83e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-6.01e4 - 6.01e4i)T + 8.58e9iT^{2} \) |
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
−11.76021658755296693971333833334, −10.50169438425168553828492093441, −9.328939166863691959301025968276, −8.849255946706480687416308185331, −7.77791224319098859628143934613, −6.23364363664962838707853897319, −4.69696917287948654460133321260, −3.81946622318301853089893582848, −2.43571397938896933413995454360, −0.24830470077930439217168376090,
1.66502583403857000031525412542, 3.13490356367081216279331344211, 4.10037302990281648563331578764, 6.47390715532336780711256656848, 6.82793887768240113294798426621, 8.328580774215637187844827680967, 8.733560718207595265576043041944, 10.66201491964647488982829459196, 11.01203300048445565557070885843, 12.51761272684607225467596499241