L(s) = 1 | + (1.52 − 2.63i)3-s + i·5-s + (2.67 − 1.54i)7-s + (−3.12 − 5.41i)9-s + (5.36 + 3.09i)11-s + (2.54 − 2.55i)13-s + (2.63 + 1.52i)15-s + (3.24 + 5.61i)17-s + (−2.65 + 1.53i)19-s − 9.40i·21-s + (−0.896 + 1.55i)23-s − 25-s − 9.88·27-s + (−1.49 + 2.58i)29-s − 4.34i·31-s + ⋯ |
L(s) = 1 | + (0.878 − 1.52i)3-s + 0.447i·5-s + (1.01 − 0.584i)7-s + (−1.04 − 1.80i)9-s + (1.61 + 0.933i)11-s + (0.704 − 0.709i)13-s + (0.680 + 0.392i)15-s + (0.786 + 1.36i)17-s + (−0.609 + 0.351i)19-s − 2.05i·21-s + (−0.186 + 0.323i)23-s − 0.200·25-s − 1.90·27-s + (−0.276 + 0.479i)29-s − 0.780i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.274 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.274 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.614862850\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.614862850\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (-2.54 + 2.55i)T \) |
good | 3 | \( 1 + (-1.52 + 2.63i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (-2.67 + 1.54i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.36 - 3.09i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.24 - 5.61i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.65 - 1.53i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.896 - 1.55i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.49 - 2.58i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4.34iT - 31T^{2} \) |
| 37 | \( 1 + (8.31 + 4.80i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.40 + 3.69i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.64 - 4.58i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 3.46iT - 47T^{2} \) |
| 53 | \( 1 + 7.82T + 53T^{2} \) |
| 59 | \( 1 + (1.21 - 0.699i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.92 + 8.52i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.52 - 0.881i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.60 + 2.08i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 11.7iT - 73T^{2} \) |
| 79 | \( 1 + 6.58T + 79T^{2} \) |
| 83 | \( 1 + 4.46iT - 83T^{2} \) |
| 89 | \( 1 + (10.0 + 5.80i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (10.6 - 6.14i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.606624383820050666028883010314, −8.576171023712391098673765465016, −8.031896612903676769809543149312, −7.33546436616692458622775000199, −6.61882369865159537353390513690, −5.81865255208916245623909767135, −4.12226730614040065981566873892, −3.39176310793822523856882196825, −1.78935255490155753565446788957, −1.42426017487673592709458939895,
1.61765450099762429672020880744, 3.06386923485453471538636510226, 3.92680079007848261807970001261, 4.71599039135103205830725293340, 5.46460055046126492576163822392, 6.67367025839025956842660921692, 8.116240937963840872967661827515, 8.777997591018293361184650488462, 9.022819988286817079450576978634, 9.869228513942590883320794558076