L(s) = 1 | + i·2-s − 0.585i·3-s − 4-s + 0.585·6-s − i·8-s + 2.65·9-s + 4.82·11-s + 0.585i·12-s − 0.828i·13-s + 16-s + 5.41i·17-s + 2.65i·18-s + 3.41·19-s + 4.82i·22-s − 6.82i·23-s − 0.585·24-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.338i·3-s − 0.5·4-s + 0.239·6-s − 0.353i·8-s + 0.885·9-s + 1.45·11-s + 0.169i·12-s − 0.229i·13-s + 0.250·16-s + 1.31i·17-s + 0.626i·18-s + 0.783·19-s + 1.02i·22-s − 1.42i·23-s − 0.119·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.055436741\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.055436741\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 0.585iT - 3T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + 0.828iT - 13T^{2} \) |
| 17 | \( 1 - 5.41iT - 17T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 + 6.82iT - 23T^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 + 3.65iT - 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 + 3.17iT - 43T^{2} \) |
| 47 | \( 1 - 10.8iT - 47T^{2} \) |
| 53 | \( 1 + 10.4iT - 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 - 9.65iT - 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 6.58iT - 73T^{2} \) |
| 79 | \( 1 - 1.17T + 79T^{2} \) |
| 83 | \( 1 + 6.24iT - 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + 16.2iT - 97T^{2} \) |
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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
−8.714715502434862605279801041594, −8.302008191466248462960313994812, −7.25182884415519037209577498759, −6.75664569861252901118475009163, −6.12140095851912650593277190011, −5.16194806951121567740626823273, −4.14717672050199002335861964486, −3.61370844262757165358118933171, −1.98928027887369660942751943713, −0.947696966719384440904817519222,
1.02931522940193074541380487008, 1.92866865948844188178278373139, 3.34574644365841410024040179447, 3.83505467024691348348920852431, 4.81380078792670869640652719493, 5.47740471991652956578594464177, 6.79218739176064516139916515313, 7.20333481540859506719701681816, 8.304688602063123866649376956342, 9.350138246118015908393462819664