L(s) = 1 | + (0.248 + 1.39i)2-s + (1.22 − 1.22i)3-s + (−1.87 + 0.691i)4-s + (0.705 − 0.407i)5-s + (2.01 + 1.39i)6-s + (2.03 + 1.69i)7-s + (−1.42 − 2.44i)8-s + (−0.0104 − 2.99i)9-s + (0.742 + 0.881i)10-s + (4.17 + 2.41i)11-s + (−1.44 + 3.14i)12-s + (−2.20 − 1.27i)13-s + (−1.85 + 3.25i)14-s + (0.363 − 1.36i)15-s + (3.04 − 2.59i)16-s + (−0.503 + 0.290i)17-s + ⋯ |
L(s) = 1 | + (0.175 + 0.984i)2-s + (0.705 − 0.708i)3-s + (−0.938 + 0.345i)4-s + (0.315 − 0.182i)5-s + (0.821 + 0.570i)6-s + (0.768 + 0.639i)7-s + (−0.505 − 0.862i)8-s + (−0.00349 − 0.999i)9-s + (0.234 + 0.278i)10-s + (1.25 + 0.727i)11-s + (−0.417 + 0.908i)12-s + (−0.611 − 0.352i)13-s + (−0.494 + 0.869i)14-s + (0.0937 − 0.352i)15-s + (0.760 − 0.649i)16-s + (−0.122 + 0.0704i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.58311 + 0.630324i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.58311 + 0.630324i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.248 - 1.39i)T \) |
| 3 | \( 1 + (-1.22 + 1.22i)T \) |
| 7 | \( 1 + (-2.03 - 1.69i)T \) |
good | 5 | \( 1 + (-0.705 + 0.407i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.17 - 2.41i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.20 + 1.27i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.503 - 0.290i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.24 - 3.88i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.60 + 3.23i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.16 + 5.47i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 9.62T + 31T^{2} \) |
| 37 | \( 1 + (-1.62 + 2.82i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.04 + 1.75i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.72 - 2.72i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 9.58T + 47T^{2} \) |
| 53 | \( 1 + (1.09 + 1.89i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 + 7.24iT - 61T^{2} \) |
| 67 | \( 1 + 5.02iT - 67T^{2} \) |
| 71 | \( 1 + 3.64iT - 71T^{2} \) |
| 73 | \( 1 + (-2.26 + 1.30i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 10.1iT - 79T^{2} \) |
| 83 | \( 1 + (0.820 + 1.42i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.90 + 1.10i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.14 - 3.54i)T + (48.5 - 84.0i)T^{2} \) |
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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
−12.55550019222914989705326141742, −11.55652639243979829234105988894, −9.676859340659734922936444988983, −9.042507862770446773223959035828, −8.136371002473016491020710506874, −7.24675431136078254042011256357, −6.27635785758002985100765476035, −5.11238739870475994622790330596, −3.76199037088527553402592614865, −1.86986796063668871620819399512,
1.77191666219599417344650095568, 3.28746783917238440500783630903, 4.29324494649521324200225600337, 5.27730633654432219037682855687, 7.05344355443065749402496550193, 8.527196299614265081110245830285, 9.170793865579632375710453112713, 10.07061638208549708638287526459, 11.09593739712045691011282871345, 11.50176484928760753144894192643