L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + 1.44·5-s − 0.999·8-s + (0.724 − 1.25i)10-s − 2·11-s + (−2.44 + 4.24i)13-s + (−0.5 + 0.866i)16-s + (1 − 1.73i)17-s + (−1.27 − 2.20i)19-s + (−0.724 − 1.25i)20-s + (−1 + 1.73i)22-s + 23-s − 2.89·25-s + (2.44 + 4.24i)26-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + 0.648·5-s − 0.353·8-s + (0.229 − 0.396i)10-s − 0.603·11-s + (−0.679 + 1.17i)13-s + (−0.125 + 0.216i)16-s + (0.242 − 0.420i)17-s + (−0.292 − 0.506i)19-s + (−0.162 − 0.280i)20-s + (−0.213 + 0.369i)22-s + 0.208·23-s − 0.579·25-s + (0.480 + 0.832i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.145297170\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.145297170\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 1.44T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + (2.44 - 4.24i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.27 + 2.20i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 + (3.44 + 5.97i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3 + 5.19i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.89 + 10.2i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.89 + 8.48i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.44 + 5.97i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.89 + 8.48i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.44 - 9.43i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1 + 1.73i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.27 - 5.67i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.44 - 11.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 0.101T + 71T^{2} \) |
| 73 | \( 1 + (-3.44 + 5.97i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.949 + 1.64i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1 - 1.73i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (8.44 + 14.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.44 + 2.51i)T + (-48.5 + 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.890050948584235168064490408216, −7.57285200219251898495012036054, −7.04777892388786478659587238627, −5.89397592893851604084834055108, −5.42549434460005154573901243632, −4.44444468565613571388869128622, −3.71067345905651718916853067654, −2.34105996004405276648102374414, −2.04495518324881043493612846358, −0.30311296919093299490525242244,
1.54593159369374135794875357111, 2.81870560113647883150505077547, 3.53402365955597775554143195896, 4.86576411869001529398445927734, 5.28767289068896530420694871625, 6.10491750947709974555001640471, 6.80508668627485757014896212253, 7.87059434293360700121410446778, 8.097197144599799957154090559601, 9.203155647610398750775359415243