L(s) = 1 | + (−1.41 + 2.44i)5-s + (−1 − 1.73i)11-s + 5.65·13-s + (−1.41 − 2.44i)17-s + (−2.82 + 4.89i)19-s + (−3 + 5.19i)23-s + (−1.49 − 2.59i)25-s − 4·29-s + (2.82 + 4.89i)31-s + (1 − 1.73i)37-s − 2.82·41-s − 4·43-s + (5.65 − 9.79i)47-s + (−6 − 10.3i)53-s + 5.65·55-s + ⋯ |
L(s) = 1 | + (−0.632 + 1.09i)5-s + (−0.301 − 0.522i)11-s + 1.56·13-s + (−0.342 − 0.594i)17-s + (−0.648 + 1.12i)19-s + (−0.625 + 1.08i)23-s + (−0.299 − 0.519i)25-s − 0.742·29-s + (0.508 + 0.879i)31-s + (0.164 − 0.284i)37-s − 0.441·41-s − 0.609·43-s + (0.825 − 1.42i)47-s + (−0.824 − 1.42i)53-s + 0.762·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6279828940\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6279828940\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1.41 - 2.44i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + (1.41 + 2.44i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.82 - 4.89i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + (-2.82 - 4.89i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2.82T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (-5.65 + 9.79i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.65 - 9.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.82 - 4.89i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6 - 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + (4.24 - 7.34i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−8.630803386858349803889810384604, −8.284128055691339425415691529969, −7.38746469539236020439903240139, −6.75582501516725827036933261681, −5.99975163820132977448402643375, −5.31958347556492921911805582203, −3.85111824381347163125414451099, −3.68088204535689104773267375777, −2.65699591385508786614902251628, −1.44915848150364844547056413159,
0.19499898755856863083878066500, 1.35756320633370239334647848808, 2.47274397015080130763037242510, 3.74858046341464022964079554106, 4.38064270842646198907727175390, 4.95553250835819972330296838879, 6.10206686275671847625346843454, 6.55950255130791717461790073068, 7.78333486263716821958233487865, 8.201868430388122185112348904984