| L(s) = 1 | − 1.91·2-s + 3-s + 1.65·4-s − 0.339·5-s − 1.91·6-s − 3.26·7-s + 0.659·8-s + 9-s + 0.648·10-s − 2.68·11-s + 1.65·12-s − 13-s + 6.23·14-s − 0.339·15-s − 4.57·16-s + 0.748·17-s − 1.91·18-s + 5.50·19-s − 0.561·20-s − 3.26·21-s + 5.13·22-s − 4.13·23-s + 0.659·24-s − 4.88·25-s + 1.91·26-s + 27-s − 5.39·28-s + ⋯ |
| L(s) = 1 | − 1.35·2-s + 0.577·3-s + 0.827·4-s − 0.151·5-s − 0.780·6-s − 1.23·7-s + 0.233·8-s + 0.333·9-s + 0.205·10-s − 0.810·11-s + 0.477·12-s − 0.277·13-s + 1.66·14-s − 0.0875·15-s − 1.14·16-s + 0.181·17-s − 0.450·18-s + 1.26·19-s − 0.125·20-s − 0.711·21-s + 1.09·22-s − 0.862·23-s + 0.134·24-s − 0.976·25-s + 0.374·26-s + 0.192·27-s − 1.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 1.91T + 2T^{2} \) |
| 5 | \( 1 + 0.339T + 5T^{2} \) |
| 7 | \( 1 + 3.26T + 7T^{2} \) |
| 11 | \( 1 + 2.68T + 11T^{2} \) |
| 17 | \( 1 - 0.748T + 17T^{2} \) |
| 19 | \( 1 - 5.50T + 19T^{2} \) |
| 23 | \( 1 + 4.13T + 23T^{2} \) |
| 29 | \( 1 - 8.25T + 29T^{2} \) |
| 31 | \( 1 - 9.77T + 31T^{2} \) |
| 37 | \( 1 - 6.92T + 37T^{2} \) |
| 41 | \( 1 - 0.230T + 41T^{2} \) |
| 43 | \( 1 + 9.67T + 43T^{2} \) |
| 47 | \( 1 + 6.78T + 47T^{2} \) |
| 53 | \( 1 + 0.371T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 - 4.52T + 71T^{2} \) |
| 73 | \( 1 + 9.39T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 + 5.28T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 + 7.29T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.155125189060010287322565952497, −7.66597988455799889912939418359, −6.87499678955324698469768658026, −6.20156841410783428117200755513, −5.07690653615507545759882189345, −4.12039407637896940177353807928, −3.06184043284802573437707004090, −2.45808726394832801955671977840, −1.13761669062371569845174585161, 0,
1.13761669062371569845174585161, 2.45808726394832801955671977840, 3.06184043284802573437707004090, 4.12039407637896940177353807928, 5.07690653615507545759882189345, 6.20156841410783428117200755513, 6.87499678955324698469768658026, 7.66597988455799889912939418359, 8.155125189060010287322565952497
