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
−8.304893750317122621317875146525, −7.62252979646997935484874528239, −6.58151504729851726811614251934, −5.98012426149774034361216759470, −5.39710458995509635389171464443, −4.40273025330957255399073033877, −3.36309532670098148611032725691, −2.91952678833599250740109641560, −1.90740085409058175520204839999, −0.008830706586360142698772395959,
1.42114997248082188888658922500, 1.68018121357819702099534236578, 3.58916621463473059549554195582, 3.99385398686453141558500985981, 4.77613519209823935185244316607, 5.72220747279880841932244337594, 6.74446379190563583573464875712, 7.13548760510835966250725198645, 7.77146203756904968256133395412, 8.727479327100585972842822690377