L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + i·8-s − 9-s − i·12-s + (1 + i)13-s + 16-s + i·18-s + i·23-s − 24-s + i·25-s + (1 − i)26-s − i·27-s + ⋯ |
L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + i·8-s − 9-s − i·12-s + (1 + i)13-s + 16-s + i·18-s + i·23-s − 24-s + i·25-s + (1 − i)26-s − i·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8963594521\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8963594521\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 23 | \( 1 - iT \) |
good | 5 | \( 1 - iT^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 13 | \( 1 + (-1 - i)T + iT^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 29 | \( 1 + (-1 + i)T - iT^{2} \) |
| 31 | \( 1 - 2iT - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + 2T + T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (-1 + i)T - iT^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 2iT - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.14557909251180529172325303703, −9.406433429271158619751818383608, −8.809020359058085170396494193831, −8.059119333395022455122198764954, −6.59592396407637288282397194931, −5.49642998074701204040540971248, −4.69730497101381016813393064313, −3.78559143736109042547736897636, −3.08760447423791260672461362255, −1.62722133825104244685436381857,
0.915504670663884908587965140384, 2.69631262354301774915516536310, 3.94191660277821711646236674385, 5.15965159723940255656643051507, 6.05148595446201581980349313352, 6.56820439090432198566923588825, 7.51207712404278987016517739194, 8.339219614178581690611389782229, 8.625888226060976460508526230496, 9.873738888630128494489335998468