L(s) = 1 | + (0.634 − 0.773i)2-s + (−0.195 − 0.980i)4-s + (0.471 + 0.881i)7-s + (−0.881 − 0.471i)8-s + (−0.290 − 0.956i)9-s + (−0.457 − 1.82i)11-s + (0.980 + 0.195i)14-s + (−0.923 + 0.382i)16-s + (−0.923 − 0.382i)18-s + (−1.70 − 0.805i)22-s + (1.21 + 1.47i)23-s + (−0.0980 − 0.995i)25-s + (0.773 − 0.634i)28-s + (−0.761 − 1.27i)29-s + (−0.290 + 0.956i)32-s + ⋯ |
L(s) = 1 | + (0.634 − 0.773i)2-s + (−0.195 − 0.980i)4-s + (0.471 + 0.881i)7-s + (−0.881 − 0.471i)8-s + (−0.290 − 0.956i)9-s + (−0.457 − 1.82i)11-s + (0.980 + 0.195i)14-s + (−0.923 + 0.382i)16-s + (−0.923 − 0.382i)18-s + (−1.70 − 0.805i)22-s + (1.21 + 1.47i)23-s + (−0.0980 − 0.995i)25-s + (0.773 − 0.634i)28-s + (−0.761 − 1.27i)29-s + (−0.290 + 0.956i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.448128270\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.448128270\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.634 + 0.773i)T \) |
| 7 | \( 1 + (-0.471 - 0.881i)T \) |
good | 3 | \( 1 + (0.290 + 0.956i)T^{2} \) |
| 5 | \( 1 + (0.0980 + 0.995i)T^{2} \) |
| 11 | \( 1 + (0.457 + 1.82i)T + (-0.881 + 0.471i)T^{2} \) |
| 13 | \( 1 + (0.995 + 0.0980i)T^{2} \) |
| 17 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 19 | \( 1 + (-0.773 + 0.634i)T^{2} \) |
| 23 | \( 1 + (-1.21 - 1.47i)T + (-0.195 + 0.980i)T^{2} \) |
| 29 | \( 1 + (0.761 + 1.27i)T + (-0.471 + 0.881i)T^{2} \) |
| 31 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 37 | \( 1 + (-0.207 - 0.439i)T + (-0.634 + 0.773i)T^{2} \) |
| 41 | \( 1 + (0.980 + 0.195i)T^{2} \) |
| 43 | \( 1 + (-1.37 + 1.02i)T + (0.290 - 0.956i)T^{2} \) |
| 47 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 53 | \( 1 + (-0.150 + 0.251i)T + (-0.471 - 0.881i)T^{2} \) |
| 59 | \( 1 + (0.995 - 0.0980i)T^{2} \) |
| 61 | \( 1 + (-0.956 + 0.290i)T^{2} \) |
| 67 | \( 1 + (-0.235 - 1.58i)T + (-0.956 + 0.290i)T^{2} \) |
| 71 | \( 1 + (1.59 + 0.482i)T + (0.831 + 0.555i)T^{2} \) |
| 73 | \( 1 + (0.555 + 0.831i)T^{2} \) |
| 79 | \( 1 + (-1.65 - 1.10i)T + (0.382 + 0.923i)T^{2} \) |
| 83 | \( 1 + (0.634 + 0.773i)T^{2} \) |
| 89 | \( 1 + (0.195 + 0.980i)T^{2} \) |
| 97 | \( 1 + (0.707 - 0.707i)T^{2} \) |
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
−9.110065835771781513075519681290, −8.781796773734692229321362125287, −7.76386113797882651866578673438, −6.40209201071776559722028997061, −5.75940330777821364304488646180, −5.29117431182318886803585066115, −4.00643999573569765156342270574, −3.19871565423191412931761375667, −2.41839525252736730875433165490, −0.916880922059763149526283814363,
1.93782750958841770822411692477, 3.04156839435188720168257839594, 4.37005622882323528286721770234, 4.76085913244221665318020255317, 5.50427937045206980544132014866, 6.76895682851479338107249088432, 7.38822058167198482927997046729, 7.75664018378767819067155651920, 8.768643395555571938991928483594, 9.595144886302802029482827641303