| L(s) = 1 | + (−0.756 + 1.19i)2-s + (−0.950 + 0.107i)3-s + (−0.856 − 1.80i)4-s + (−1.46 + 3.04i)5-s + (0.590 − 1.21i)6-s + (−2.38 − 1.90i)7-s + (2.80 + 0.342i)8-s + (−2.03 + 0.464i)9-s + (−2.52 − 4.05i)10-s + (−2.73 + 1.72i)11-s + (1.00 + 1.62i)12-s + (2.10 + 0.480i)13-s + (4.08 − 1.41i)14-s + (1.06 − 3.04i)15-s + (−2.53 + 3.09i)16-s + (0.808 − 0.808i)17-s + ⋯ |
| L(s) = 1 | + (−0.534 + 0.845i)2-s + (−0.548 + 0.0618i)3-s + (−0.428 − 0.903i)4-s + (−0.655 + 1.36i)5-s + (0.241 − 0.496i)6-s + (−0.902 − 0.719i)7-s + (0.992 + 0.120i)8-s + (−0.677 + 0.154i)9-s + (−0.799 − 1.28i)10-s + (−0.825 + 0.518i)11-s + (0.290 + 0.469i)12-s + (0.583 + 0.133i)13-s + (1.09 − 0.378i)14-s + (0.275 − 0.787i)15-s + (−0.632 + 0.774i)16-s + (0.196 − 0.196i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0192368 - 0.294454i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0192368 - 0.294454i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.756 - 1.19i)T \) |
| 29 | \( 1 + (-0.900 - 5.30i)T \) |
| good | 3 | \( 1 + (0.950 - 0.107i)T + (2.92 - 0.667i)T^{2} \) |
| 5 | \( 1 + (1.46 - 3.04i)T + (-3.11 - 3.90i)T^{2} \) |
| 7 | \( 1 + (2.38 + 1.90i)T + (1.55 + 6.82i)T^{2} \) |
| 11 | \( 1 + (2.73 - 1.72i)T + (4.77 - 9.91i)T^{2} \) |
| 13 | \( 1 + (-2.10 - 0.480i)T + (11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (-0.808 + 0.808i)T - 17iT^{2} \) |
| 19 | \( 1 + (0.220 - 1.95i)T + (-18.5 - 4.22i)T^{2} \) |
| 23 | \( 1 + (-1.95 - 4.05i)T + (-14.3 + 17.9i)T^{2} \) |
| 31 | \( 1 + (-2.96 - 8.48i)T + (-24.2 + 19.3i)T^{2} \) |
| 37 | \( 1 + (0.187 - 0.297i)T + (-16.0 - 33.3i)T^{2} \) |
| 41 | \( 1 + (7.20 + 7.20i)T + 41iT^{2} \) |
| 43 | \( 1 + (5.16 + 1.80i)T + (33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (6.75 + 10.7i)T + (-20.3 + 42.3i)T^{2} \) |
| 53 | \( 1 + (-8.91 - 4.29i)T + (33.0 + 41.4i)T^{2} \) |
| 59 | \( 1 - 7.94iT - 59T^{2} \) |
| 61 | \( 1 + (0.705 + 6.25i)T + (-59.4 + 13.5i)T^{2} \) |
| 67 | \( 1 + (0.0163 + 0.0715i)T + (-60.3 + 29.0i)T^{2} \) |
| 71 | \( 1 + (3.44 - 15.0i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (5.92 + 2.07i)T + (57.0 + 45.5i)T^{2} \) |
| 79 | \( 1 + (0.396 - 0.631i)T + (-34.2 - 71.1i)T^{2} \) |
| 83 | \( 1 + (-2.93 + 2.34i)T + (18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (4.83 - 1.69i)T + (69.5 - 55.4i)T^{2} \) |
| 97 | \( 1 + (0.831 - 7.38i)T + (-94.5 - 21.5i)T^{2} \) |
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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
−14.27171463743866357877061995479, −13.38431491011925770573469717543, −11.74600615335372682402106834238, −10.53152300192336725206386069293, −10.26103240928751544127997813754, −8.534051386739718163928335383859, −7.22357645268922449555251075136, −6.68396694361686306444182182774, −5.31473727111387443512150763301, −3.40773592521517997522888737142,
0.39923846770453334729626394301, 3.03120346015676592675819929574, 4.70962729544346490716732636465, 6.07268808885264789162626460903, 8.132421823694341727638339708613, 8.692819095301348852033005163450, 9.801627827857564574168815045463, 11.19694194784987896649864911605, 11.89258936003169862861629149096, 12.79990874775217401291905381260
