| L(s) = 1 | + (−0.321 − 0.356i)2-s + (0.520 − 1.60i)3-s + (0.184 − 1.75i)4-s + (−3.28 + 1.46i)5-s + (−0.738 + 0.328i)6-s + (3.82 + 0.812i)7-s + (−1.46 + 1.06i)8-s + (0.135 + 0.0981i)9-s + (1.57 + 0.701i)10-s + 2.82·11-s + (−2.72 − 1.21i)12-s + (−0.937 + 1.62i)13-s + (−0.937 − 1.62i)14-s + (0.632 + 6.01i)15-s + (−2.61 − 0.555i)16-s + (−0.0185 + 0.176i)17-s + ⋯ |
| L(s) = 1 | + (−0.227 − 0.252i)2-s + (0.300 − 0.924i)3-s + (0.0924 − 0.879i)4-s + (−1.46 + 0.653i)5-s + (−0.301 + 0.134i)6-s + (1.44 + 0.306i)7-s + (−0.517 + 0.376i)8-s + (0.0450 + 0.0327i)9-s + (0.498 + 0.221i)10-s + 0.850·11-s + (−0.785 − 0.349i)12-s + (−0.259 + 0.450i)13-s + (−0.250 − 0.433i)14-s + (0.163 + 1.55i)15-s + (−0.653 − 0.138i)16-s + (−0.00450 + 0.0428i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.677337 - 0.442962i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.677337 - 0.442962i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 61 | \( 1 + (1.04 + 7.74i)T \) |
| good | 2 | \( 1 + (0.321 + 0.356i)T + (-0.209 + 1.98i)T^{2} \) |
| 3 | \( 1 + (-0.520 + 1.60i)T + (-2.42 - 1.76i)T^{2} \) |
| 5 | \( 1 + (3.28 - 1.46i)T + (3.34 - 3.71i)T^{2} \) |
| 7 | \( 1 + (-3.82 - 0.812i)T + (6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + (0.937 - 1.62i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.0185 - 0.176i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (7.01 - 1.49i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (0.839 + 0.610i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (2.12 + 3.67i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.17 - 3.52i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (-2.17 - 6.69i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.96 + 6.05i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (0.486 + 4.62i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (-5.16 - 8.94i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.649 - 0.471i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (4.50 + 5.00i)T + (-6.16 + 58.6i)T^{2} \) |
| 67 | \( 1 + (-0.492 + 0.219i)T + (44.8 - 49.7i)T^{2} \) |
| 71 | \( 1 + (0.836 + 0.372i)T + (47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (2.96 + 1.32i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (0.0647 + 0.616i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-0.957 - 1.06i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (-3.69 + 11.3i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (0.129 - 0.143i)T + (-10.1 - 96.4i)T^{2} \) |
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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
−14.72114669827611389301739379944, −14.13972602256238147401089014640, −12.27102974735049557924452500393, −11.47991778213383295098254593256, −10.64063734861331643821071066321, −8.702394863775483985332382762436, −7.72091123868422122004523527614, −6.55248300905928148856042761270, −4.44326473000852106780977142220, −1.91442023474209230060847232247,
3.87932685217849450355233056997, 4.52605738733242302685290999471, 7.29671154815016974861350571935, 8.277281782321020720933509820675, 9.023308357950637186805274095834, 10.91093694006328288635787387350, 11.81035253292831270423893180139, 12.81570637845982516120659850364, 14.76141961229394409942623110484, 15.22447500867689551488798261209
