L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (2.54 + 0.732i)7-s − 0.999i·8-s + (−2.07 − 1.19i)11-s − 5.67i·13-s + (2.56 − 0.636i)14-s + (−0.5 − 0.866i)16-s + (1.03 − 1.79i)17-s + (5.12 − 2.95i)19-s − 2.39·22-s + (−1.61 + 0.930i)23-s + (−2.83 − 4.91i)26-s + (1.90 − 1.83i)28-s + 4.88i·29-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.960 + 0.276i)7-s − 0.353i·8-s + (−0.625 − 0.361i)11-s − 1.57i·13-s + (0.686 − 0.170i)14-s + (−0.125 − 0.216i)16-s + (0.251 − 0.435i)17-s + (1.17 − 0.678i)19-s − 0.511·22-s + (−0.336 + 0.194i)23-s + (−0.556 − 0.964i)26-s + (0.360 − 0.346i)28-s + 0.907i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.243 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.243 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.669144735\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.669144735\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.54 - 0.732i)T \) |
good | 11 | \( 1 + (2.07 + 1.19i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5.67iT - 13T^{2} \) |
| 17 | \( 1 + (-1.03 + 1.79i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.12 + 2.95i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.61 - 0.930i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 4.88iT - 29T^{2} \) |
| 31 | \( 1 + (3.92 + 2.26i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.48 + 2.57i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 7.04T + 41T^{2} \) |
| 43 | \( 1 + 8.55T + 43T^{2} \) |
| 47 | \( 1 + (-2.78 - 4.83i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.62 - 2.09i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.00 + 1.73i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-10.7 + 6.22i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.81 + 6.60i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 9.14iT - 71T^{2} \) |
| 73 | \( 1 + (-0.937 - 0.541i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8.38 + 14.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + (6.63 + 11.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 12.8iT - 97T^{2} \) |
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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
−8.337778089363556945339748170290, −7.73929881891813023295951310361, −7.04534500192929282240745522704, −5.81808482208973892159195643551, −5.25888276888700612676348029200, −4.87067096593239173871153377310, −3.50290161468922640500976839917, −2.95513206797602051516924581770, −1.87595200226847290003265567695, −0.66030571127381596757857383841,
1.49371142724001314562929900589, 2.31317466904119124914827851423, 3.64293977939607707029519921011, 4.25128987973253914389989431037, 5.13528484254781010980274135335, 5.64401380686280350361400107589, 6.82940391752716474290792284346, 7.19972090585861137335745923384, 8.167418679839434742469712330386, 8.550106481091779268067068346635