L(s) = 1 | − 2.35·2-s + 1.62i·3-s + 3.53·4-s + 0.259i·5-s − 3.81i·6-s + 1.04i·7-s − 3.62·8-s + 0.370·9-s − 0.609i·10-s + 3.29i·11-s + 5.74i·12-s − 0.845·13-s − 2.45i·14-s − 0.420·15-s + 1.45·16-s + (−3.66 − 1.88i)17-s + ⋯ |
L(s) = 1 | − 1.66·2-s + 0.936i·3-s + 1.76·4-s + 0.115i·5-s − 1.55i·6-s + 0.393i·7-s − 1.28·8-s + 0.123·9-s − 0.192i·10-s + 0.992i·11-s + 1.65i·12-s − 0.234·13-s − 0.654i·14-s − 0.108·15-s + 0.362·16-s + (−0.889 − 0.456i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.124953 + 0.517684i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.124953 + 0.517684i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (3.66 + 1.88i)T \) |
| 43 | \( 1 - T \) |
good | 2 | \( 1 + 2.35T + 2T^{2} \) |
| 3 | \( 1 - 1.62iT - 3T^{2} \) |
| 5 | \( 1 - 0.259iT - 5T^{2} \) |
| 7 | \( 1 - 1.04iT - 7T^{2} \) |
| 11 | \( 1 - 3.29iT - 11T^{2} \) |
| 13 | \( 1 + 0.845T + 13T^{2} \) |
| 19 | \( 1 - 2.70T + 19T^{2} \) |
| 23 | \( 1 - 3.45iT - 23T^{2} \) |
| 29 | \( 1 - 6.17iT - 29T^{2} \) |
| 31 | \( 1 - 4.24iT - 31T^{2} \) |
| 37 | \( 1 + 7.67iT - 37T^{2} \) |
| 41 | \( 1 + 3.87iT - 41T^{2} \) |
| 47 | \( 1 + 3.53T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 + 1.58T + 59T^{2} \) |
| 61 | \( 1 - 4.40iT - 61T^{2} \) |
| 67 | \( 1 + 9.54T + 67T^{2} \) |
| 71 | \( 1 - 13.8iT - 71T^{2} \) |
| 73 | \( 1 - 3.06iT - 73T^{2} \) |
| 79 | \( 1 - 0.391iT - 79T^{2} \) |
| 83 | \( 1 + 1.93T + 83T^{2} \) |
| 89 | \( 1 + 8.91T + 89T^{2} \) |
| 97 | \( 1 + 1.22iT - 97T^{2} \) |
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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
−10.57152545254787680210182880458, −9.641302322249447332268018272434, −9.287203881654179209102244357058, −8.538174740976510484480939615370, −7.25452777480787888064535050712, −6.97406728130243424831618328633, −5.35977406863603520044852892201, −4.38803268723795618038583264410, −2.89493898644271803658587150916, −1.59192081137552467630835805363,
0.49841252444025977713456655285, 1.56137411813804967185042831173, 2.78940930065575605282209033737, 4.50310275153154578880241746965, 6.22583808694428656002843437480, 6.73103722118204441384240026037, 7.74404802123517104866228092818, 8.181756106095100851277342228309, 9.076290571874441859131191904585, 9.913108517747582982210905849182