L(s) = 1 | − 2.63i·2-s − 4.93·4-s + 1.79i·7-s + 7.73i·8-s + 1.80·11-s + 1.97i·13-s + 4.73·14-s + 10.4·16-s − 4.80i·17-s − 2.96·19-s − 4.76i·22-s − 1.73i·23-s + 5.19·26-s − 8.87i·28-s + 7.36·29-s + ⋯ |
L(s) = 1 | − 1.86i·2-s − 2.46·4-s + 0.679i·7-s + 2.73i·8-s + 0.545·11-s + 0.546i·13-s + 1.26·14-s + 2.62·16-s − 1.16i·17-s − 0.680·19-s − 1.01i·22-s − 0.361i·23-s + 1.01·26-s − 1.67i·28-s + 1.36·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.044625796\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.044625796\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 2.63iT - 2T^{2} \) |
| 7 | \( 1 - 1.79iT - 7T^{2} \) |
| 11 | \( 1 - 1.80T + 11T^{2} \) |
| 13 | \( 1 - 1.97iT - 13T^{2} \) |
| 17 | \( 1 + 4.80iT - 17T^{2} \) |
| 19 | \( 1 + 2.96T + 19T^{2} \) |
| 23 | \( 1 + 1.73iT - 23T^{2} \) |
| 29 | \( 1 - 7.36T + 29T^{2} \) |
| 31 | \( 1 + 2.62T + 31T^{2} \) |
| 37 | \( 1 + 11.6iT - 37T^{2} \) |
| 41 | \( 1 + 2.46T + 41T^{2} \) |
| 43 | \( 1 + 7.27iT - 43T^{2} \) |
| 47 | \( 1 + 6.29iT - 47T^{2} \) |
| 53 | \( 1 - 1.72iT - 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 9.10iT - 67T^{2} \) |
| 71 | \( 1 - 1.27T + 71T^{2} \) |
| 73 | \( 1 + 3.58iT - 73T^{2} \) |
| 79 | \( 1 + 2.11T + 79T^{2} \) |
| 83 | \( 1 - 1.09iT - 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 3.83iT - 97T^{2} \) |
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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
−9.103427155744517579577795087346, −8.442361147908037910746983459542, −7.23953079458372765115103253581, −6.11509494671069215999066192522, −5.06108919461153251157084590245, −4.37575561070682940959708402838, −3.47278880662361088884504973645, −2.53200072005461911125476286815, −1.81171299347363335788793195324, −0.42769262030207582986946232876,
1.20091999660649536534909648677, 3.29680302217456312313001041021, 4.29537733061498390332381056362, 4.83277693843911881567078170177, 6.02709166143540766807432708749, 6.36537439409205779240734560638, 7.20020733686352957527244154619, 7.988698792180639581169843830214, 8.445662986407929651388134160517, 9.274551173281803838282029428588