| L(s) = 1 | + 1.55·2-s + 0.467·3-s + 0.410·4-s − 1.37·5-s + 0.726·6-s − 3.35·7-s − 2.46·8-s − 2.78·9-s − 2.13·10-s + 2.05·11-s + 0.192·12-s − 1.60·13-s − 5.20·14-s − 0.644·15-s − 4.65·16-s − 17-s − 4.31·18-s + 4.43·19-s − 0.564·20-s − 1.56·21-s + 3.19·22-s − 2.95·23-s − 1.15·24-s − 3.10·25-s − 2.49·26-s − 2.70·27-s − 1.37·28-s + ⋯ |
| L(s) = 1 | + 1.09·2-s + 0.270·3-s + 0.205·4-s − 0.615·5-s + 0.296·6-s − 1.26·7-s − 0.872·8-s − 0.927·9-s − 0.675·10-s + 0.620·11-s + 0.0554·12-s − 0.445·13-s − 1.39·14-s − 0.166·15-s − 1.16·16-s − 0.242·17-s − 1.01·18-s + 1.01·19-s − 0.126·20-s − 0.342·21-s + 0.681·22-s − 0.616·23-s − 0.235·24-s − 0.621·25-s − 0.488·26-s − 0.520·27-s − 0.259·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 1.55T + 2T^{2} \) |
| 3 | \( 1 - 0.467T + 3T^{2} \) |
| 5 | \( 1 + 1.37T + 5T^{2} \) |
| 7 | \( 1 + 3.35T + 7T^{2} \) |
| 11 | \( 1 - 2.05T + 11T^{2} \) |
| 13 | \( 1 + 1.60T + 13T^{2} \) |
| 19 | \( 1 - 4.43T + 19T^{2} \) |
| 23 | \( 1 + 2.95T + 23T^{2} \) |
| 29 | \( 1 + 3.13T + 29T^{2} \) |
| 31 | \( 1 + 4.72T + 31T^{2} \) |
| 37 | \( 1 - 6.73T + 37T^{2} \) |
| 41 | \( 1 - 9.28T + 41T^{2} \) |
| 47 | \( 1 + 8.52T + 47T^{2} \) |
| 53 | \( 1 - 9.33T + 53T^{2} \) |
| 59 | \( 1 + 9.83T + 59T^{2} \) |
| 61 | \( 1 - 2.75T + 61T^{2} \) |
| 67 | \( 1 - 8.84T + 67T^{2} \) |
| 71 | \( 1 + 1.37T + 71T^{2} \) |
| 73 | \( 1 + 0.237T + 73T^{2} \) |
| 79 | \( 1 + 7.25T + 79T^{2} \) |
| 83 | \( 1 + 2.15T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−9.636356672510607432848327470142, −9.334888056856925858937426460117, −8.213159074108972597148798098600, −7.16382076688123639464718917725, −6.14386475681703177786822373728, −5.50115242423230845791444184146, −4.17760811363505970694803797222, −3.50560523527321925559032479873, −2.65902646729624576157635952031, 0,
2.65902646729624576157635952031, 3.50560523527321925559032479873, 4.17760811363505970694803797222, 5.50115242423230845791444184146, 6.14386475681703177786822373728, 7.16382076688123639464718917725, 8.213159074108972597148798098600, 9.334888056856925858937426460117, 9.636356672510607432848327470142
