| L(s) = 1 | + (1.34 + 1.74i)2-s + (1.37 + 1.05i)3-s + (−0.738 + 2.75i)4-s + (−1.77 − 0.116i)5-s + (0.0106 + 3.81i)6-s + (−0.295 − 4.50i)7-s + (−1.73 + 0.718i)8-s + (0.792 + 2.89i)9-s + (−2.17 − 3.25i)10-s + (−3.64 − 3.19i)11-s + (−3.91 + 3.01i)12-s + (3.24 + 0.868i)13-s + (7.47 − 6.55i)14-s + (−2.31 − 2.02i)15-s + (1.36 + 0.785i)16-s + (−1.37 + 3.88i)17-s + ⋯ |
| L(s) = 1 | + (0.948 + 1.23i)2-s + (0.795 + 0.606i)3-s + (−0.369 + 1.37i)4-s + (−0.792 − 0.0519i)5-s + (0.00433 + 1.55i)6-s + (−0.111 − 1.70i)7-s + (−0.612 + 0.253i)8-s + (0.264 + 0.964i)9-s + (−0.686 − 1.02i)10-s + (−1.09 − 0.962i)11-s + (−1.12 + 0.871i)12-s + (0.899 + 0.240i)13-s + (1.99 − 1.75i)14-s + (−0.598 − 0.521i)15-s + (0.340 + 0.196i)16-s + (−0.332 + 0.942i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.154 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.23798 + 1.44649i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23798 + 1.44649i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.37 - 1.05i)T \) |
| 17 | \( 1 + (1.37 - 3.88i)T \) |
| good | 2 | \( 1 + (-1.34 - 1.74i)T + (-0.517 + 1.93i)T^{2} \) |
| 5 | \( 1 + (1.77 + 0.116i)T + (4.95 + 0.652i)T^{2} \) |
| 7 | \( 1 + (0.295 + 4.50i)T + (-6.94 + 0.913i)T^{2} \) |
| 11 | \( 1 + (3.64 + 3.19i)T + (1.43 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-3.24 - 0.868i)T + (11.2 + 6.5i)T^{2} \) |
| 19 | \( 1 + (2.09 + 0.867i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-1.20 - 3.54i)T + (-18.2 + 14.0i)T^{2} \) |
| 29 | \( 1 + (3.39 - 1.67i)T + (17.6 - 23.0i)T^{2} \) |
| 31 | \( 1 + (0.656 + 0.748i)T + (-4.04 + 30.7i)T^{2} \) |
| 37 | \( 1 + (-0.0324 - 0.163i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-1.80 + 3.66i)T + (-24.9 - 32.5i)T^{2} \) |
| 43 | \( 1 + (-1.31 + 9.95i)T + (-41.5 - 11.1i)T^{2} \) |
| 47 | \( 1 + (-7.98 + 2.13i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-1.75 + 4.24i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-2.19 - 1.68i)T + (15.2 + 56.9i)T^{2} \) |
| 61 | \( 1 + (-7.16 + 0.469i)T + (60.4 - 7.96i)T^{2} \) |
| 67 | \( 1 + (0.191 - 0.110i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.68 - 0.931i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-3.14 - 2.10i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (6.37 - 7.26i)T + (-10.3 - 78.3i)T^{2} \) |
| 83 | \( 1 + (1.15 - 0.886i)T + (21.4 - 80.1i)T^{2} \) |
| 89 | \( 1 + (9.41 - 9.41i)T - 89iT^{2} \) |
| 97 | \( 1 + (-6.36 - 12.9i)T + (-59.0 + 76.9i)T^{2} \) |
| 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
−13.49819384752753600786005489267, −13.04630675268674702309887578602, −11.02190294252189638541301632727, −10.41320055907150208388370585670, −8.590631675176297603400771034803, −7.83663769183486422010195616555, −7.03078174294476721286792599627, −5.50528668157711457566894745328, −4.01819634696692819216193209024, −3.73317585192419688665786205735,
2.20892965925012977300477370840, 3.02502752899459150251628265434, 4.47820235988990608119167343966, 5.87007600634914116647081719768, 7.58153242640271900547254047530, 8.626902147060216702763740201097, 9.759043581014818113832103424343, 11.15572210614106358119433290575, 11.98659294123619127387173580164, 12.70831129901527673612169031250
