L(s) = 1 | − 0.441·2-s − 1.59·3-s − 1.80·4-s − 1.49·5-s + 0.705·6-s − 0.804·7-s + 1.67·8-s − 0.443·9-s + 0.661·10-s + 5.43·11-s + 2.88·12-s + 3.66·13-s + 0.355·14-s + 2.39·15-s + 2.86·16-s − 17-s + 0.195·18-s + 1.59·19-s + 2.70·20-s + 1.28·21-s − 2.39·22-s − 7.99·23-s − 2.68·24-s − 2.75·25-s − 1.61·26-s + 5.50·27-s + 1.45·28-s + ⋯ |
L(s) = 1 | − 0.312·2-s − 0.923·3-s − 0.902·4-s − 0.669·5-s + 0.288·6-s − 0.304·7-s + 0.593·8-s − 0.147·9-s + 0.209·10-s + 1.63·11-s + 0.833·12-s + 1.01·13-s + 0.0949·14-s + 0.618·15-s + 0.717·16-s − 0.242·17-s + 0.0461·18-s + 0.366·19-s + 0.604·20-s + 0.280·21-s − 0.511·22-s − 1.66·23-s − 0.548·24-s − 0.551·25-s − 0.316·26-s + 1.05·27-s + 0.274·28-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 | \( 1 + 0.441T + 2T^{2} \) |
| 3 | \( 1 + 1.59T + 3T^{2} \) |
| 5 | \( 1 + 1.49T + 5T^{2} \) |
| 7 | \( 1 + 0.804T + 7T^{2} \) |
| 11 | \( 1 - 5.43T + 11T^{2} \) |
| 13 | \( 1 - 3.66T + 13T^{2} \) |
| 19 | \( 1 - 1.59T + 19T^{2} \) |
| 23 | \( 1 + 7.99T + 23T^{2} \) |
| 29 | \( 1 - 8.81T + 29T^{2} \) |
| 31 | \( 1 + 1.91T + 31T^{2} \) |
| 37 | \( 1 + 6.23T + 37T^{2} \) |
| 41 | \( 1 + 8.52T + 41T^{2} \) |
| 43 | \( 1 - 0.104T + 43T^{2} \) |
| 47 | \( 1 + 7.54T + 47T^{2} \) |
| 53 | \( 1 - 7.27T + 53T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 - 7.57T + 71T^{2} \) |
| 73 | \( 1 - 6.24T + 73T^{2} \) |
| 79 | \( 1 + 4.70T + 79T^{2} \) |
| 83 | \( 1 + 1.71T + 83T^{2} \) |
| 89 | \( 1 - 0.392T + 89T^{2} \) |
| 97 | \( 1 + 8.37T + 97T^{2} \) |
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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
−9.613390791046154327238420950618, −8.577679700244767840916329888496, −8.235491402629982686972778767376, −6.82409254337833881173735700484, −6.19793821061156616693615958219, −5.21265914088675902889526574194, −4.13217951196280171810790498307, −3.56220547326943846408639079948, −1.31989134926767978957438345135, 0,
1.31989134926767978957438345135, 3.56220547326943846408639079948, 4.13217951196280171810790498307, 5.21265914088675902889526574194, 6.19793821061156616693615958219, 6.82409254337833881173735700484, 8.235491402629982686972778767376, 8.577679700244767840916329888496, 9.613390791046154327238420950618