L(s) = 1 | − i·3-s + (−0.504 − 2.17i)5-s − 3.91i·7-s − 9-s + 3.46·11-s + 6.46i·13-s + (−2.17 + 0.504i)15-s − 0.603i·17-s − 0.547·19-s − 3.91·21-s + 7.27i·23-s + (−4.49 + 2.19i)25-s + i·27-s − 5.89·29-s − 4.98·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (−0.225 − 0.974i)5-s − 1.48i·7-s − 0.333·9-s + 1.04·11-s + 1.79i·13-s + (−0.562 + 0.130i)15-s − 0.146i·17-s − 0.125·19-s − 0.854·21-s + 1.51i·23-s + (−0.898 + 0.439i)25-s + 0.192i·27-s − 1.09·29-s − 0.895·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.03206185606\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03206185606\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (0.504 + 2.17i)T \) |
| 67 | \( 1 + iT \) |
good | 7 | \( 1 + 3.91iT - 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 6.46iT - 13T^{2} \) |
| 17 | \( 1 + 0.603iT - 17T^{2} \) |
| 19 | \( 1 + 0.547T + 19T^{2} \) |
| 23 | \( 1 - 7.27iT - 23T^{2} \) |
| 29 | \( 1 + 5.89T + 29T^{2} \) |
| 31 | \( 1 + 4.98T + 31T^{2} \) |
| 37 | \( 1 - 2.50iT - 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 3.38iT - 43T^{2} \) |
| 47 | \( 1 + 5.74iT - 47T^{2} \) |
| 53 | \( 1 - 2.06iT - 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 + 8.87T + 61T^{2} \) |
| 71 | \( 1 - 3.83T + 71T^{2} \) |
| 73 | \( 1 - 15.6iT - 73T^{2} \) |
| 79 | \( 1 + 8.63T + 79T^{2} \) |
| 83 | \( 1 + 3.96iT - 83T^{2} \) |
| 89 | \( 1 + 8.55T + 89T^{2} \) |
| 97 | \( 1 + 6.10iT - 97T^{2} \) |
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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
−8.727479327100585972842822690377, −7.77146203756904968256133395412, −7.13548760510835966250725198645, −6.74446379190563583573464875712, −5.72220747279880841932244337594, −4.77613519209823935185244316607, −3.99385398686453141558500985981, −3.58916621463473059549554195582, −1.68018121357819702099534236578, −1.42114997248082188888658922500,
0.008830706586360142698772395959, 1.90740085409058175520204839999, 2.91952678833599250740109641560, 3.36309532670098148611032725691, 4.40273025330957255399073033877, 5.39710458995509635389171464443, 5.98012426149774034361216759470, 6.58151504729851726811614251934, 7.62252979646997935484874528239, 8.304893750317122621317875146525