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 | \( 1 + (0.707 + 0.707i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.595144886302802029482827641303, −8.768643395555571938991928483594, −7.75664018378767819067155651920, −7.38822058167198482927997046729, −6.76895682851479338107249088432, −5.50427937045206980544132014866, −4.76085913244221665318020255317, −4.37005622882323528286721770234, −3.04156839435188720168257839594, −1.93782750958841770822411692477,
0.916880922059763149526283814363, 2.41839525252736730875433165490, 3.19871565423191412931761375667, 4.00643999573569765156342270574, 5.29117431182318886803585066115, 5.75940330777821364304488646180, 6.40209201071776559722028997061, 7.76386113797882651866578673438, 8.781796773734692229321362125287, 9.110065835771781513075519681290