| L(s) = 1 | − 1.64·2-s + 3-s + 0.692·4-s + 0.0817·5-s − 1.64·6-s + 2.14·8-s + 9-s − 0.134·10-s − 4.94·11-s + 0.692·12-s + 1.23·13-s + 0.0817·15-s − 4.90·16-s − 0.810·17-s − 1.64·18-s + 6.55·19-s + 0.0566·20-s + 8.11·22-s − 5.53·23-s + 2.14·24-s − 4.99·25-s − 2.02·26-s + 27-s − 0.0447·29-s − 0.134·30-s + 7.20·31-s + 3.75·32-s + ⋯ |
| L(s) = 1 | − 1.16·2-s + 0.577·3-s + 0.346·4-s + 0.0365·5-s − 0.669·6-s + 0.758·8-s + 0.333·9-s − 0.0424·10-s − 1.49·11-s + 0.199·12-s + 0.341·13-s + 0.0211·15-s − 1.22·16-s − 0.196·17-s − 0.386·18-s + 1.50·19-s + 0.0126·20-s + 1.72·22-s − 1.15·23-s + 0.437·24-s − 0.998·25-s − 0.396·26-s + 0.192·27-s − 0.00830·29-s − 0.0244·30-s + 1.29·31-s + 0.664·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 1.64T + 2T^{2} \) |
| 5 | \( 1 - 0.0817T + 5T^{2} \) |
| 11 | \( 1 + 4.94T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 + 0.810T + 17T^{2} \) |
| 19 | \( 1 - 6.55T + 19T^{2} \) |
| 23 | \( 1 + 5.53T + 23T^{2} \) |
| 29 | \( 1 + 0.0447T + 29T^{2} \) |
| 31 | \( 1 - 7.20T + 31T^{2} \) |
| 37 | \( 1 - 5.22T + 37T^{2} \) |
| 43 | \( 1 + 8.64T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 5.99T + 53T^{2} \) |
| 59 | \( 1 + 4.34T + 59T^{2} \) |
| 61 | \( 1 - 1.13T + 61T^{2} \) |
| 67 | \( 1 + 2.70T + 67T^{2} \) |
| 71 | \( 1 + 5.56T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 - 9.02T + 83T^{2} \) |
| 89 | \( 1 - 7.38T + 89T^{2} \) |
| 97 | \( 1 - 5.44T + 97T^{2} \) |
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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
−7.85199505949303421605554788739, −7.54597959886073528951687197056, −6.51719138437183906319772182567, −5.60462312094952095523264393046, −4.81793459562122241311099604428, −3.98878942321070535952147864138, −2.96520705918886601581184867912, −2.18859080783189454466413911988, −1.19611426158060910040759906020, 0,
1.19611426158060910040759906020, 2.18859080783189454466413911988, 2.96520705918886601581184867912, 3.98878942321070535952147864138, 4.81793459562122241311099604428, 5.60462312094952095523264393046, 6.51719138437183906319772182567, 7.54597959886073528951687197056, 7.85199505949303421605554788739
