L(s) = 1 | − 3.46·7-s + 6.92i·13-s + 8i·19-s + 5·25-s − 10.3·31-s + 6.92i·37-s − 8i·43-s + 4.99·49-s − 6.92i·61-s + 16i·67-s − 10·73-s + 17.3·79-s − 23.9i·91-s − 14·97-s + 3.46·103-s + ⋯ |
L(s) = 1 | − 1.30·7-s + 1.92i·13-s + 1.83i·19-s + 25-s − 1.86·31-s + 1.13i·37-s − 1.21i·43-s + 0.714·49-s − 0.887i·61-s + 1.95i·67-s − 1.17·73-s + 1.94·79-s − 2.51i·91-s − 1.42·97-s + 0.341·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.531857 + 0.693130i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.531857 + 0.693130i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 6.92iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 8iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 - 6.92iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 16iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 17.3T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−10.92360165800917855017185872965, −9.965066290601168683592973602306, −9.328342953022103823308423458819, −8.501974515438193711159163136782, −7.17200878518640598048076687065, −6.56056507341188421033187019462, −5.60454693503504828223059983383, −4.19970158031309724699419925657, −3.34173539391058334856183077115, −1.81762917176668932327044231594,
0.48177600340166529814338386913, 2.71705476216629341101441829728, 3.46881006421485080198859346656, 4.96571226389503390515010892979, 5.89352805678870864365951215710, 6.87858262977726324973536900542, 7.69312328104009617759245389385, 8.905868698652704995489566248196, 9.519175689722656529341327941221, 10.55487254875962330697560427659