L(s) = 1 | + (0.900 + 1.56i)3-s + (−0.5 + 0.866i)4-s − 1.24·5-s + (−1.12 + 1.94i)9-s + (0.5 + 0.866i)11-s − 1.80·12-s + (−1.12 − 1.94i)15-s + (−0.499 − 0.866i)16-s + (0.623 − 1.07i)20-s + (0.222 + 0.385i)23-s + 0.554·25-s − 2.24·27-s + 0.445·31-s + (−0.900 + 1.56i)33-s + (−1.12 − 1.94i)36-s + (−0.900 − 1.56i)37-s + ⋯ |
L(s) = 1 | + (0.900 + 1.56i)3-s + (−0.5 + 0.866i)4-s − 1.24·5-s + (−1.12 + 1.94i)9-s + (0.5 + 0.866i)11-s − 1.80·12-s + (−1.12 − 1.94i)15-s + (−0.499 − 0.866i)16-s + (0.623 − 1.07i)20-s + (0.222 + 0.385i)23-s + 0.554·25-s − 2.24·27-s + 0.445·31-s + (−0.900 + 1.56i)33-s + (−1.12 − 1.94i)36-s + (−0.900 − 1.56i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9216763726\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9216763726\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 3 | \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + 1.24T + T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - 0.445T + T^{2} \) |
| 37 | \( 1 + (0.900 + 1.56i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 - 0.445T + T^{2} \) |
| 53 | \( 1 - 1.24T + T^{2} \) |
| 59 | \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.623 + 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.727822088236610749029578723580, −8.778549665347243781739913178650, −8.658120345024665781057957104518, −7.61114929080680926971866250053, −7.17866930828601063426633275304, −5.38985382303255524494021876246, −4.47331552860339295384790494166, −4.01785958866756952172148102533, −3.48594262308510766057951267835, −2.49860207756640481287262127317,
0.63888356687725366145910625436, 1.68083895241522822737107859156, 3.01778991815372593514134567190, 3.79894262483043957234214970432, 4.91764685008333796867960319237, 6.19630690280850177477578155025, 6.63786991534231890075153757742, 7.60697896588974570630626085706, 8.243071878100391837260322544211, 8.744590157231603032968287111309