| L(s) = 1 | + i·2-s − 4-s − 1.63i·5-s + (1.91 + 1.82i)7-s − i·8-s + 1.63·10-s − 2.23·11-s − 0.801·13-s + (−1.82 + 1.91i)14-s + 16-s + 2.71i·17-s + (−3.13 − 3.02i)19-s + 1.63i·20-s − 2.23i·22-s − 6.49·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.733i·5-s + (0.722 + 0.691i)7-s − 0.353i·8-s + 0.518·10-s − 0.672·11-s − 0.222·13-s + (−0.488 + 0.511i)14-s + 0.250·16-s + 0.658i·17-s + (−0.719 − 0.694i)19-s + 0.366i·20-s − 0.475i·22-s − 1.35·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0394i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5761801103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5761801103\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.91 - 1.82i)T \) |
| 19 | \( 1 + (3.13 + 3.02i)T \) |
| good | 5 | \( 1 + 1.63iT - 5T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 + 0.801T + 13T^{2} \) |
| 17 | \( 1 - 2.71iT - 17T^{2} \) |
| 23 | \( 1 + 6.49T + 23T^{2} \) |
| 29 | \( 1 - 3.82iT - 29T^{2} \) |
| 31 | \( 1 + 5.19T + 31T^{2} \) |
| 37 | \( 1 - 7.19iT - 37T^{2} \) |
| 41 | \( 1 + 6.99T + 41T^{2} \) |
| 43 | \( 1 - 7.13T + 43T^{2} \) |
| 47 | \( 1 + 1.91iT - 47T^{2} \) |
| 53 | \( 1 + 1.19iT - 53T^{2} \) |
| 59 | \( 1 + 4.59T + 59T^{2} \) |
| 61 | \( 1 - 0.934iT - 61T^{2} \) |
| 67 | \( 1 - 15.2iT - 67T^{2} \) |
| 71 | \( 1 + 1.19iT - 71T^{2} \) |
| 73 | \( 1 - 6.99iT - 73T^{2} \) |
| 79 | \( 1 - 0.503iT - 79T^{2} \) |
| 83 | \( 1 + 11.1iT - 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 + 17.9T + 97T^{2} \) |
| 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
−8.973871556064301201505028040797, −8.539739249079367464547635329734, −7.978853656298172921298226230828, −7.10605236929869271565366266899, −6.14957968350392585236883919103, −5.38233073142852914733552661953, −4.83201033601081385042706304196, −3.99916515995845128038154919453, −2.61134988662428552670022064564, −1.49762839541618508898275198898,
0.18721158587498686545345567448, 1.76290981369980060354930895113, 2.57557968580832640921997408127, 3.65235987586724653648131671855, 4.37354785339448654653339171153, 5.28763194646855208638677895700, 6.19190920357609301451497703808, 7.26080955520405940871931474010, 7.79311380060968641546126975593, 8.543098556482108584463640813057
