L(s) = 1 | + (−1.25 − 2.17i)5-s + (1.71 + 2.01i)7-s + (−0.542 + 0.939i)11-s + (−0.457 + 0.792i)13-s + (−1.39 − 2.41i)17-s + (0.667 − 1.15i)19-s + (−3.06 − 5.30i)23-s + (−0.640 + 1.10i)25-s + (0.414 + 0.718i)29-s + 2.61·31-s + (2.23 − 6.24i)35-s + (3.14 − 5.44i)37-s + (0.817 − 1.41i)41-s + (0.795 + 1.37i)43-s + 6.59·47-s + ⋯ |
L(s) = 1 | + (−0.560 − 0.970i)5-s + (0.646 + 0.762i)7-s + (−0.163 + 0.283i)11-s + (−0.126 + 0.219i)13-s + (−0.337 − 0.585i)17-s + (0.153 − 0.265i)19-s + (−0.638 − 1.10i)23-s + (−0.128 + 0.221i)25-s + (0.0769 + 0.133i)29-s + 0.469·31-s + (0.378 − 1.05i)35-s + (0.517 − 0.895i)37-s + (0.127 − 0.221i)41-s + (0.121 + 0.210i)43-s + 0.961·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.106 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.255780386\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.255780386\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.71 - 2.01i)T \) |
good | 5 | \( 1 + (1.25 + 2.17i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.542 - 0.939i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.457 - 0.792i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.39 + 2.41i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.667 + 1.15i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.06 + 5.30i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.414 - 0.718i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.61T + 31T^{2} \) |
| 37 | \( 1 + (-3.14 + 5.44i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.817 + 1.41i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.795 - 1.37i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 6.59T + 47T^{2} \) |
| 53 | \( 1 + (3.95 + 6.85i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 5.07T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 - 9.64T + 71T^{2} \) |
| 73 | \( 1 + (5.03 + 8.72i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 9.34T + 79T^{2} \) |
| 83 | \( 1 + (4.09 + 7.09i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.36 + 11.0i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.46 + 14.6i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.797969936850282185572301607739, −8.135386705497139667627448529370, −7.47684804161104798904241756292, −6.44281081864031086914903618360, −5.50913766702828319387278471008, −4.68960155218166196072270326240, −4.27488065448380125083322550888, −2.82649544800781749307218446829, −1.87214234157232524726099838967, −0.46345655423000142410485629197,
1.24863137524508205091808167725, 2.57613757866788938472460471510, 3.57461935842176899438377717784, 4.21299770908857912217352451088, 5.26130242941579813794094627048, 6.23273848909636835019494647876, 6.99286304287904932685029079774, 7.83348242441819974398000243504, 8.070437121830983830332036560274, 9.273576246747962652131557707694