L(s) = 1 | + 0.153·2-s − 0.702·3-s − 1.97·4-s − 3.12·5-s − 0.107·6-s + 0.325·7-s − 0.610·8-s − 2.50·9-s − 0.479·10-s + 4.04·11-s + 1.38·12-s − 1.61·13-s + 0.0499·14-s + 2.19·15-s + 3.85·16-s + 4.90·17-s − 0.384·18-s − 3.65·19-s + 6.18·20-s − 0.228·21-s + 0.619·22-s + 6.16·23-s + 0.428·24-s + 4.78·25-s − 0.248·26-s + 3.86·27-s − 0.643·28-s + ⋯ |
L(s) = 1 | + 0.108·2-s − 0.405·3-s − 0.988·4-s − 1.39·5-s − 0.0440·6-s + 0.123·7-s − 0.215·8-s − 0.835·9-s − 0.151·10-s + 1.21·11-s + 0.401·12-s − 0.448·13-s + 0.0133·14-s + 0.567·15-s + 0.964·16-s + 1.19·17-s − 0.0906·18-s − 0.839·19-s + 1.38·20-s − 0.0499·21-s + 0.132·22-s + 1.28·23-s + 0.0875·24-s + 0.956·25-s − 0.0486·26-s + 0.744·27-s − 0.121·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2671 | \( 1 + T \) |
good | 2 | \( 1 - 0.153T + 2T^{2} \) |
| 3 | \( 1 + 0.702T + 3T^{2} \) |
| 5 | \( 1 + 3.12T + 5T^{2} \) |
| 7 | \( 1 - 0.325T + 7T^{2} \) |
| 11 | \( 1 - 4.04T + 11T^{2} \) |
| 13 | \( 1 + 1.61T + 13T^{2} \) |
| 17 | \( 1 - 4.90T + 17T^{2} \) |
| 19 | \( 1 + 3.65T + 19T^{2} \) |
| 23 | \( 1 - 6.16T + 23T^{2} \) |
| 29 | \( 1 + 2.85T + 29T^{2} \) |
| 31 | \( 1 - 4.20T + 31T^{2} \) |
| 37 | \( 1 - 1.46T + 37T^{2} \) |
| 41 | \( 1 - 2.60T + 41T^{2} \) |
| 43 | \( 1 + 4.96T + 43T^{2} \) |
| 47 | \( 1 + 2.30T + 47T^{2} \) |
| 53 | \( 1 + 2.74T + 53T^{2} \) |
| 59 | \( 1 + 5.29T + 59T^{2} \) |
| 61 | \( 1 - 8.12T + 61T^{2} \) |
| 67 | \( 1 - 4.83T + 67T^{2} \) |
| 71 | \( 1 + 6.56T + 71T^{2} \) |
| 73 | \( 1 + 5.18T + 73T^{2} \) |
| 79 | \( 1 + 4.15T + 79T^{2} \) |
| 83 | \( 1 - 5.48T + 83T^{2} \) |
| 89 | \( 1 - 2.76T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−8.403867429182858223122422303023, −7.912986599508275680438251881444, −6.97297742443149146392445725811, −6.09917799825610308010135183641, −5.15887827939142733755701881372, −4.51139561177132181553223106209, −3.72231975716706129503865748663, −3.03739869338979661610934887839, −1.09812774036760587190494433385, 0,
1.09812774036760587190494433385, 3.03739869338979661610934887839, 3.72231975716706129503865748663, 4.51139561177132181553223106209, 5.15887827939142733755701881372, 6.09917799825610308010135183641, 6.97297742443149146392445725811, 7.912986599508275680438251881444, 8.403867429182858223122422303023