| L(s) = 1 | − 3-s + 5-s − i·7-s − i·11-s − 15-s + i·21-s + 25-s + 27-s + (1 − i)31-s + i·33-s − i·35-s − i·37-s − i·41-s + (−1 + i)43-s + i·47-s + ⋯ |
| L(s) = 1 | − 3-s + 5-s − i·7-s − i·11-s − 15-s + i·21-s + 25-s + 27-s + (1 − i)31-s + i·33-s − i·35-s − i·37-s − i·41-s + (−1 + i)43-s + i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7803770009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7803770009\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + iT \) |
| good | 3 | \( 1 + T + T^{2} \) |
| 7 | \( 1 + iT - T^{2} \) |
| 11 | \( 1 + iT - T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 + (-1 + i)T - iT^{2} \) |
| 41 | \( 1 + iT - T^{2} \) |
| 43 | \( 1 + (1 - i)T - iT^{2} \) |
| 47 | \( 1 - iT - T^{2} \) |
| 53 | \( 1 - iT - T^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (1 - i)T - iT^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + (-1 - i)T + iT^{2} \) |
| 83 | \( 1 - iT - T^{2} \) |
| 89 | \( 1 + (1 - i)T - iT^{2} \) |
| 97 | \( 1 + (-1 + i)T - iT^{2} \) |
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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
−10.67516636882129806210701009476, −9.799286297463342101753300914211, −8.867974839052776531917143822463, −7.79575930962859574121482498576, −6.66478762581296740870360209884, −6.02213689077084962837281446081, −5.28798658406050219517083468675, −4.17011262634207891587118467124, −2.77139635544225333279026596199, −1.02748974401997445720603175672,
1.76449077401087519533179252045, 2.93856423426109703182556366949, 4.81288738167812688382663982630, 5.30374524985244088228701352809, 6.26784537709608327171777563953, 6.81138345199920516422076649275, 8.269091570514110769839769188153, 9.103515677188854980741298460547, 9.994769794085147461570745753155, 10.56862508602636943109469905733
