| L(s) = 1 | + (0.809 + 0.587i)2-s + (0.104 + 0.994i)3-s + (0.309 + 0.951i)4-s + (0.0864 + 0.149i)5-s + (−0.5 + 0.866i)6-s + (1.43 + 0.305i)7-s + (−0.309 + 0.951i)8-s + (−0.978 + 0.207i)9-s + (−0.0180 + 0.171i)10-s + (−0.615 + 0.683i)11-s + (−0.913 + 0.406i)12-s + (0.151 + 0.0672i)13-s + (0.981 + 1.09i)14-s + (−0.139 + 0.101i)15-s + (−0.809 + 0.587i)16-s + (−2.08 − 2.31i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (0.0603 + 0.574i)3-s + (0.154 + 0.475i)4-s + (0.0386 + 0.0669i)5-s + (−0.204 + 0.353i)6-s + (0.542 + 0.115i)7-s + (−0.109 + 0.336i)8-s + (−0.326 + 0.0693i)9-s + (−0.00571 + 0.0543i)10-s + (−0.185 + 0.206i)11-s + (−0.263 + 0.117i)12-s + (0.0418 + 0.0186i)13-s + (0.262 + 0.291i)14-s + (−0.0361 + 0.0262i)15-s + (−0.202 + 0.146i)16-s + (−0.506 − 0.562i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30692 + 0.952472i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30692 + 0.952472i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 3 | \( 1 + (-0.104 - 0.994i)T \) |
| 31 | \( 1 + (-5.44 + 1.17i)T \) |
| good | 5 | \( 1 + (-0.0864 - 0.149i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.43 - 0.305i)T + (6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 + (0.615 - 0.683i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.151 - 0.0672i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (2.08 + 2.31i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.62 + 0.725i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-0.671 + 2.06i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (5.06 + 3.68i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (0.100 - 0.174i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.0842 - 0.801i)T + (-40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (-5.23 + 2.33i)T + (28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (2.48 - 1.80i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (10.0 - 2.14i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.392 - 3.73i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 - 0.625T + 61T^{2} \) |
| 67 | \( 1 + (-2.69 - 4.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.93 + 0.624i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (7.89 - 8.77i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (4.30 + 4.78i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (1.27 - 12.1i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (2.00 + 6.16i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-3.15 - 9.69i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−12.91939391418880718471455696919, −11.79170874063421141882311198519, −10.99172707208655550265295402988, −9.807208193162519866251330412232, −8.678939301752813838019376284193, −7.65348045284212885457953604028, −6.39641860166515526396646632122, −5.14198283887679220105909834534, −4.24661767534882264644221776066, −2.65147618744717448056439007834,
1.62482153391418539184216863486, 3.26628030373551873702500965356, 4.78778201372337834264097252436, 5.93354138818552952607862383682, 7.16223132427309010086137793395, 8.276250060447940748490703631339, 9.460830069109602726606806989328, 10.80857125122131050358126595847, 11.44009072973893129152858670781, 12.55330187898339542244502515615
