| L(s) = 1 | + (−1.42 + 1.19i)2-s + (0.215 − 0.0188i)3-s + (0.252 − 1.43i)4-s + (−1.89 − 1.18i)5-s + (−0.284 + 0.284i)6-s + (1.45 − 3.11i)7-s + (−0.508 − 0.880i)8-s + (−2.90 + 0.512i)9-s + (4.11 − 0.588i)10-s + (0.652 − 0.376i)11-s + (0.0273 − 0.313i)12-s + (0.682 − 3.87i)13-s + (1.65 + 6.16i)14-s + (−0.431 − 0.218i)15-s + (4.50 + 1.63i)16-s + (−2.88 + 0.508i)17-s + ⋯ |
| L(s) = 1 | + (−1.00 + 0.844i)2-s + (0.124 − 0.0108i)3-s + (0.126 − 0.715i)4-s + (−0.849 − 0.527i)5-s + (−0.116 + 0.116i)6-s + (0.548 − 1.17i)7-s + (−0.179 − 0.311i)8-s + (−0.969 + 0.170i)9-s + (1.30 − 0.186i)10-s + (0.196 − 0.113i)11-s + (0.00790 − 0.0903i)12-s + (0.189 − 1.07i)13-s + (0.441 + 1.64i)14-s + (−0.111 − 0.0564i)15-s + (1.12 + 0.409i)16-s + (−0.699 + 0.123i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.552 + 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.414444 - 0.222417i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.414444 - 0.222417i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.89 + 1.18i)T \) |
| 37 | \( 1 + (1.70 + 5.83i)T \) |
| good | 2 | \( 1 + (1.42 - 1.19i)T + (0.347 - 1.96i)T^{2} \) |
| 3 | \( 1 + (-0.215 + 0.0188i)T + (2.95 - 0.520i)T^{2} \) |
| 7 | \( 1 + (-1.45 + 3.11i)T + (-4.49 - 5.36i)T^{2} \) |
| 11 | \( 1 + (-0.652 + 0.376i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.682 + 3.87i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (2.88 - 0.508i)T + (15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (0.0559 + 0.639i)T + (-18.7 + 3.29i)T^{2} \) |
| 23 | \( 1 + (-3.42 + 5.93i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.170 - 0.0457i)T + (25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (-0.582 - 0.582i)T + 31iT^{2} \) |
| 41 | \( 1 + (5.70 + 1.00i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + 12.7T + 43T^{2} \) |
| 47 | \( 1 + (-1.59 - 5.94i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.06 - 4.43i)T + (-34.0 + 40.6i)T^{2} \) |
| 59 | \( 1 + (1.69 + 3.64i)T + (-37.9 + 45.1i)T^{2} \) |
| 61 | \( 1 + (-2.18 + 1.52i)T + (20.8 - 57.3i)T^{2} \) |
| 67 | \( 1 + (-11.1 - 5.21i)T + (43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (5.50 + 4.61i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-11.1 + 11.1i)T - 73iT^{2} \) |
| 79 | \( 1 + (-11.0 - 5.13i)T + (50.7 + 60.5i)T^{2} \) |
| 83 | \( 1 + (-1.09 + 1.56i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (-11.7 + 5.49i)T + (57.2 - 68.1i)T^{2} \) |
| 97 | \( 1 + (7.95 + 4.59i)T + (48.5 + 84.0i)T^{2} \) |
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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
−12.46219693085030554015946815390, −11.17110494773368324348947671237, −10.44962841575975911000497179342, −8.959653090230792670089069569440, −8.303820012758134954842727830845, −7.63442259027482115297417711442, −6.56811130317416173881384178441, −4.98149486710423417469880667062, −3.54147819590037332011411969957, −0.57701004793544425987972466073,
2.07134810335490739088920639350, 3.34267185914689023769310999257, 5.21573385240101561682883836689, 6.71861696208600692217318870246, 8.261790353931355710215167097214, 8.726789092615654032384427202789, 9.685125042818122608602069794035, 11.06333756375176390581518883373, 11.64199662875793668637303156243, 11.97861254347127253420037993345
