L(s) = 1 | + (0.433 + 1.90i)3-s + (0.222 + 0.974i)5-s + (−0.781 + 0.623i)7-s + (−2.52 + 1.21i)9-s + (−1.75 + 0.846i)15-s + (−1.52 − 1.21i)21-s + (1.21 + 1.52i)23-s + (−0.900 + 0.433i)25-s + (−2.19 − 2.74i)27-s + (0.777 − 0.974i)29-s + (−0.781 − 0.623i)35-s + (−0.400 − 1.75i)41-s + (−0.193 + 0.846i)43-s + (−1.74 − 2.19i)45-s + (1.75 + 0.846i)47-s + ⋯ |
L(s) = 1 | + (0.433 + 1.90i)3-s + (0.222 + 0.974i)5-s + (−0.781 + 0.623i)7-s + (−2.52 + 1.21i)9-s + (−1.75 + 0.846i)15-s + (−1.52 − 1.21i)21-s + (1.21 + 1.52i)23-s + (−0.900 + 0.433i)25-s + (−2.19 − 2.74i)27-s + (0.777 − 0.974i)29-s + (−0.781 − 0.623i)35-s + (−0.400 − 1.75i)41-s + (−0.193 + 0.846i)43-s + (−1.74 − 2.19i)45-s + (1.75 + 0.846i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.187747261\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.187747261\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.222 - 0.974i)T \) |
| 7 | \( 1 + (0.781 - 0.623i)T \) |
good | 3 | \( 1 + (-0.433 - 1.90i)T + (-0.900 + 0.433i)T^{2} \) |
| 11 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 13 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 17 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-1.21 - 1.52i)T + (-0.222 + 0.974i)T^{2} \) |
| 29 | \( 1 + (-0.777 + 0.974i)T + (-0.222 - 0.974i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 41 | \( 1 + (0.400 + 1.75i)T + (-0.900 + 0.433i)T^{2} \) |
| 43 | \( 1 + (0.193 - 0.846i)T + (-0.900 - 0.433i)T^{2} \) |
| 47 | \( 1 + (-1.75 - 0.846i)T + (0.623 + 0.781i)T^{2} \) |
| 53 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 59 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 61 | \( 1 + (-0.277 + 0.347i)T + (-0.222 - 0.974i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 73 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (1.40 - 0.678i)T + (0.623 - 0.781i)T^{2} \) |
| 89 | \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.237078791577098669756952702281, −8.675562799346074125927678486676, −7.72687917885534092205283152873, −6.80566935612661316720298900340, −5.78145497362691389230473722439, −5.45546750520389070534741316881, −4.37496500033167268075868887424, −3.58483212811489721679398290532, −3.00488348049055897733700081457, −2.35937186726681832952291105771,
0.64213025503338328105935771964, 1.39978115467448808826116664724, 2.51631666806888305192361402659, 3.22702869584993106621764856469, 4.42571956995619178640721325768, 5.46053046228323221238223257881, 6.24834653336806892376305838222, 6.89625005054062465799574707191, 7.31083164057488724900308386215, 8.372998510740013633613761396717