L(s) = 1 | + (0.707 − 0.707i)3-s + (1.71 + 2.01i)7-s − 1.00i·9-s − 4.27·11-s + (−2.31 + 2.31i)13-s + (−1.41 − 1.41i)17-s + 6.50·19-s + (2.63 + 0.209i)21-s + (3.37 + 3.37i)23-s + (−0.707 − 0.707i)27-s + 8.27i·29-s − 3.04i·31-s + (−3.02 + 3.02i)33-s + (7.68 − 7.68i)37-s + 3.27i·39-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (0.648 + 0.760i)7-s − 0.333i·9-s − 1.28·11-s + (−0.642 + 0.642i)13-s + (−0.342 − 0.342i)17-s + 1.49·19-s + (0.575 + 0.0456i)21-s + (0.704 + 0.704i)23-s + (−0.136 − 0.136i)27-s + 1.53i·29-s − 0.546i·31-s + (−0.526 + 0.526i)33-s + (1.26 − 1.26i)37-s + 0.524i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.857853358\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.857853358\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.71 - 2.01i)T \) |
good | 11 | \( 1 + 4.27T + 11T^{2} \) |
| 13 | \( 1 + (2.31 - 2.31i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.41 + 1.41i)T + 17iT^{2} \) |
| 19 | \( 1 - 6.50T + 19T^{2} \) |
| 23 | \( 1 + (-3.37 - 3.37i)T + 23iT^{2} \) |
| 29 | \( 1 - 8.27iT - 29T^{2} \) |
| 31 | \( 1 + 3.04iT - 31T^{2} \) |
| 37 | \( 1 + (-7.68 + 7.68i)T - 37iT^{2} \) |
| 41 | \( 1 - 12.1iT - 41T^{2} \) |
| 43 | \( 1 + (-5.53 - 5.53i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.41 + 1.41i)T + 47iT^{2} \) |
| 53 | \( 1 + (-8.61 - 8.61i)T + 53iT^{2} \) |
| 59 | \( 1 + 8.71T + 59T^{2} \) |
| 61 | \( 1 - 3.04iT - 61T^{2} \) |
| 67 | \( 1 + (3.08 - 3.08i)T - 67iT^{2} \) |
| 71 | \( 1 - 1.72T + 71T^{2} \) |
| 73 | \( 1 + (-7.07 + 7.07i)T - 73iT^{2} \) |
| 79 | \( 1 - 10.8iT - 79T^{2} \) |
| 83 | \( 1 + (-1.02 + 1.02i)T - 83iT^{2} \) |
| 89 | \( 1 + 7.88T + 89T^{2} \) |
| 97 | \( 1 + (1.92 + 1.92i)T + 97iT^{2} \) |
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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
−9.322257895084440052492412135335, −8.352734194518134732800106060953, −7.56703933445834379156864386628, −7.22247738803382568636894193352, −5.93113659330687508421567033178, −5.22697864164434599913446756123, −4.50457039359947996814408056163, −3.02551938720205145247017950829, −2.48454410660864679128077540556, −1.28141827937439763245567251564,
0.66104830716655646479321570534, 2.24930681751509527960628746859, 3.07756275440069210848893837813, 4.14792840870599123459828085364, 4.98225364829551361500081062874, 5.53599759659198348952164718664, 6.86000694042695324315425046774, 7.74273256573321051279285355541, 7.989837642104455948583234501269, 8.987957145802736362574337114110