L(s) = 1 | + 3-s − 4i·7-s − 2·9-s − 3i·11-s − i·17-s − 7i·19-s − 4i·21-s + 4i·23-s − 5·27-s + 8i·29-s + 4·31-s − 3i·33-s − 4·37-s + 3·41-s − 8·43-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.51i·7-s − 0.666·9-s − 0.904i·11-s − 0.242i·17-s − 1.60i·19-s − 0.872i·21-s + 0.834i·23-s − 0.962·27-s + 1.48i·29-s + 0.718·31-s − 0.522i·33-s − 0.657·37-s + 0.468·41-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.109402541\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.109402541\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + iT - 17T^{2} \) |
| 19 | \( 1 + 7iT - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 8iT - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 - 8iT - 59T^{2} \) |
| 61 | \( 1 - 4iT - 61T^{2} \) |
| 67 | \( 1 - 9T + 67T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 + 11iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + T + 83T^{2} \) |
| 89 | \( 1 + 13T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−8.468323593157476187460836922501, −7.48904166211153461211905980389, −7.06131983411534643152493473078, −6.15036726561459343238310876851, −5.16825691585519940113257088253, −4.37948113771519708562207378527, −3.32670329954205519514977744063, −2.96276888779833845617795639777, −1.45588449903174021487327293218, −0.29715567679261336141009170021,
1.81464293586862136607624154828, 2.44537855772120569958870141478, 3.30373326282026763388137525611, 4.32428036316403344048825060858, 5.28842577470225479057146565050, 5.97761365514370637849146937989, 6.60678698787886229234535438840, 7.912797010012268664280487930204, 8.201719620958858160263906062570, 8.894963425889064803651097854305