| L(s) = 1 | + (−0.984 + 0.173i)2-s + (−0.766 + 0.642i)3-s + (0.939 − 0.342i)4-s + (0.355 − 0.977i)5-s + (0.642 − 0.766i)6-s + (0.798 + 2.52i)7-s + (−0.866 + 0.5i)8-s + (0.173 − 0.984i)9-s + (−0.180 + 1.02i)10-s + (0.382 + 0.661i)11-s + (−0.499 + 0.866i)12-s + (3.01 + 2.52i)13-s + (−1.22 − 2.34i)14-s + (0.355 + 0.977i)15-s + (0.766 − 0.642i)16-s + (−6.83 + 1.20i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 + 0.122i)2-s + (−0.442 + 0.371i)3-s + (0.469 − 0.171i)4-s + (0.159 − 0.437i)5-s + (0.262 − 0.312i)6-s + (0.301 + 0.953i)7-s + (−0.306 + 0.176i)8-s + (0.0578 − 0.328i)9-s + (−0.0571 + 0.323i)10-s + (0.115 + 0.199i)11-s + (−0.144 + 0.249i)12-s + (0.836 + 0.701i)13-s + (−0.327 − 0.626i)14-s + (0.0918 + 0.252i)15-s + (0.191 − 0.160i)16-s + (−1.65 + 0.292i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.562242 + 0.651567i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.562242 + 0.651567i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.984 - 0.173i)T \) |
| 3 | \( 1 + (0.766 - 0.642i)T \) |
| 7 | \( 1 + (-0.798 - 2.52i)T \) |
| 19 | \( 1 + (3.75 + 2.21i)T \) |
| good | 5 | \( 1 + (-0.355 + 0.977i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (-0.382 - 0.661i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.01 - 2.52i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (6.83 - 1.20i)T + (15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-8.01 + 2.91i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.04 - 0.185i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.145 + 0.251i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 11.2iT - 37T^{2} \) |
| 41 | \( 1 + (6.99 - 5.86i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.44 - 0.890i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.26 - 0.222i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-0.669 - 1.84i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (0.607 + 3.44i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-2.67 - 7.33i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (13.9 + 2.46i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (1.51 - 4.15i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-4.07 - 4.85i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-9.22 - 10.9i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.36 - 0.789i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (5.93 + 4.98i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-1.82 - 10.3i)T + (-91.1 + 33.1i)T^{2} \) |
| 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
−10.61176725876275352757268606091, −9.374653815469600362040272482745, −8.833424756146259909073972927609, −8.405846344267308292479788830377, −6.75378361088169358012920340267, −6.40867707032796831939553369139, −5.12171154060032328557317207343, −4.42915087443969003389982918598, −2.72989573410422135089721764514, −1.42827453972258268362511759891,
0.60555015756582526570062584160, 1.98222252202736736094081504470, 3.37252459787412925622106228699, 4.59037509951224133900118102736, 5.88999032489836898980660054612, 6.80170212999844592822225281321, 7.31271096788089040705661520509, 8.415674880360490505139526503961, 9.068522318491142214461861924239, 10.37479301931868205349283929314
