L(s) = 1 | + 1.43·2-s + 0.506·3-s + 0.0686·4-s − 2.94·5-s + 0.729·6-s + 4.23·7-s − 2.77·8-s − 2.74·9-s − 4.23·10-s − 2.79·11-s + 0.0348·12-s − 4.43·13-s + 6.09·14-s − 1.49·15-s − 4.13·16-s − 17-s − 3.94·18-s + 4.64·19-s − 0.202·20-s + 2.14·21-s − 4.02·22-s − 0.575·23-s − 1.40·24-s + 3.67·25-s − 6.38·26-s − 2.91·27-s + 0.290·28-s + ⋯ |
L(s) = 1 | + 1.01·2-s + 0.292·3-s + 0.0343·4-s − 1.31·5-s + 0.297·6-s + 1.60·7-s − 0.982·8-s − 0.914·9-s − 1.33·10-s − 0.843·11-s + 0.0100·12-s − 1.23·13-s + 1.62·14-s − 0.385·15-s − 1.03·16-s − 0.242·17-s − 0.929·18-s + 1.06·19-s − 0.0452·20-s + 0.468·21-s − 0.857·22-s − 0.120·23-s − 0.287·24-s + 0.734·25-s − 1.25·26-s − 0.560·27-s + 0.0549·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 2 | \( 1 - 1.43T + 2T^{2} \) |
| 3 | \( 1 - 0.506T + 3T^{2} \) |
| 5 | \( 1 + 2.94T + 5T^{2} \) |
| 7 | \( 1 - 4.23T + 7T^{2} \) |
| 11 | \( 1 + 2.79T + 11T^{2} \) |
| 13 | \( 1 + 4.43T + 13T^{2} \) |
| 19 | \( 1 - 4.64T + 19T^{2} \) |
| 23 | \( 1 + 0.575T + 23T^{2} \) |
| 29 | \( 1 + 6.67T + 29T^{2} \) |
| 31 | \( 1 + 6.38T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 4.30T + 43T^{2} \) |
| 47 | \( 1 + 7.29T + 47T^{2} \) |
| 53 | \( 1 - 9.90T + 53T^{2} \) |
| 61 | \( 1 + 1.47T + 61T^{2} \) |
| 67 | \( 1 + 9.61T + 67T^{2} \) |
| 71 | \( 1 + 4.34T + 71T^{2} \) |
| 73 | \( 1 - 9.75T + 73T^{2} \) |
| 79 | \( 1 - 9.53T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 - 5.44T + 89T^{2} \) |
| 97 | \( 1 - 2.05T + 97T^{2} \) |
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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
−9.339597519800459407585016614933, −8.573942489707818760629577705330, −7.72322209018098679467476152489, −7.37071876025290224536563981949, −5.54313936363828153108469031376, −5.14684547267412109987095912480, −4.27170824847378135976752471664, −3.37961220958255780124994386898, −2.31172377475338789196650775125, 0,
2.31172377475338789196650775125, 3.37961220958255780124994386898, 4.27170824847378135976752471664, 5.14684547267412109987095912480, 5.54313936363828153108469031376, 7.37071876025290224536563981949, 7.72322209018098679467476152489, 8.573942489707818760629577705330, 9.339597519800459407585016614933