| L(s) = 1 | − 2-s − 3-s + 4-s + 1.62·5-s + 6-s − 4.68·7-s − 8-s + 9-s − 1.62·10-s + 3.50·11-s − 12-s + 0.306·13-s + 4.68·14-s − 1.62·15-s + 16-s − 17-s − 18-s + 7.98·19-s + 1.62·20-s + 4.68·21-s − 3.50·22-s − 0.450·23-s + 24-s − 2.35·25-s − 0.306·26-s − 27-s − 4.68·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.726·5-s + 0.408·6-s − 1.76·7-s − 0.353·8-s + 0.333·9-s − 0.513·10-s + 1.05·11-s − 0.288·12-s + 0.0850·13-s + 1.25·14-s − 0.419·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s + 1.83·19-s + 0.363·20-s + 1.02·21-s − 0.746·22-s − 0.0938·23-s + 0.204·24-s − 0.471·25-s − 0.0601·26-s − 0.192·27-s − 0.884·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 - 1.62T + 5T^{2} \) |
| 7 | \( 1 + 4.68T + 7T^{2} \) |
| 11 | \( 1 - 3.50T + 11T^{2} \) |
| 13 | \( 1 - 0.306T + 13T^{2} \) |
| 19 | \( 1 - 7.98T + 19T^{2} \) |
| 23 | \( 1 + 0.450T + 23T^{2} \) |
| 29 | \( 1 + 2.74T + 29T^{2} \) |
| 31 | \( 1 + 9.09T + 31T^{2} \) |
| 37 | \( 1 + 0.508T + 37T^{2} \) |
| 41 | \( 1 + 3.01T + 41T^{2} \) |
| 43 | \( 1 - 5.17T + 43T^{2} \) |
| 47 | \( 1 + 13.6T + 47T^{2} \) |
| 53 | \( 1 - 1.19T + 53T^{2} \) |
| 61 | \( 1 - 5.58T + 61T^{2} \) |
| 67 | \( 1 - 2.15T + 67T^{2} \) |
| 71 | \( 1 - 1.51T + 71T^{2} \) |
| 73 | \( 1 + 3.09T + 73T^{2} \) |
| 79 | \( 1 + 1.76T + 79T^{2} \) |
| 83 | \( 1 + 5.70T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 - 17.0T + 97T^{2} \) |
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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
−7.44009683665585695691541429853, −7.04910238859798080776272458568, −6.24345402356419354457668367709, −5.92494930330109242038983872584, −5.10644812892264367110057791720, −3.72550089195274383841732921721, −3.30816358649020711506640847402, −2.11524561955648693979121330871, −1.14823963676105760798332385551, 0,
1.14823963676105760798332385551, 2.11524561955648693979121330871, 3.30816358649020711506640847402, 3.72550089195274383841732921721, 5.10644812892264367110057791720, 5.92494930330109242038983872584, 6.24345402356419354457668367709, 7.04910238859798080776272458568, 7.44009683665585695691541429853
