| L(s) = 1 | + (1.99 + 1.61i)2-s + (0.238 + 0.971i)3-s + (0.970 + 4.50i)4-s + (−1.36 + 2.08i)5-s + (−1.09 + 2.32i)6-s + (2.07 − 0.540i)7-s + (−3.01 + 5.97i)8-s + (−0.886 + 0.462i)9-s + (−6.10 + 1.96i)10-s + (−1.91 − 2.75i)11-s + (−4.14 + 2.01i)12-s + (−0.583 − 0.559i)13-s + (5.02 + 2.27i)14-s + (−2.35 − 0.830i)15-s + (−7.28 + 3.29i)16-s + (1.95 − 0.166i)17-s + ⋯ |
| L(s) = 1 | + (1.41 + 1.14i)2-s + (0.137 + 0.560i)3-s + (0.485 + 2.25i)4-s + (−0.611 + 0.932i)5-s + (−0.445 + 0.949i)6-s + (0.784 − 0.204i)7-s + (−1.06 + 2.11i)8-s + (−0.295 + 0.154i)9-s + (−1.92 + 0.620i)10-s + (−0.578 − 0.829i)11-s + (−1.19 + 0.581i)12-s + (−0.161 − 0.155i)13-s + (1.34 + 0.606i)14-s + (−0.607 − 0.214i)15-s + (−1.82 + 0.823i)16-s + (0.474 − 0.0404i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.405539 + 3.09205i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.405539 + 3.09205i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.238 - 0.971i)T \) |
| 223 | \( 1 + (-14.5 + 3.16i)T \) |
| good | 2 | \( 1 + (-1.99 - 1.61i)T + (0.421 + 1.95i)T^{2} \) |
| 5 | \( 1 + (1.36 - 2.08i)T + (-1.99 - 4.58i)T^{2} \) |
| 7 | \( 1 + (-2.07 + 0.540i)T + (6.11 - 3.41i)T^{2} \) |
| 11 | \( 1 + (1.91 + 2.75i)T + (-3.81 + 10.3i)T^{2} \) |
| 13 | \( 1 + (0.583 + 0.559i)T + (0.551 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-1.95 + 0.166i)T + (16.7 - 2.87i)T^{2} \) |
| 19 | \( 1 + (-6.30 + 3.76i)T + (9.03 - 16.7i)T^{2} \) |
| 23 | \( 1 + (2.78 + 0.157i)T + (22.8 + 2.59i)T^{2} \) |
| 29 | \( 1 + (-6.00 - 4.31i)T + (9.27 + 27.4i)T^{2} \) |
| 31 | \( 1 + (1.82 - 2.20i)T + (-5.67 - 30.4i)T^{2} \) |
| 37 | \( 1 + (2.02 + 4.32i)T + (-23.6 + 28.4i)T^{2} \) |
| 41 | \( 1 + (-1.88 - 4.68i)T + (-29.6 + 28.3i)T^{2} \) |
| 43 | \( 1 + (-2.45 + 0.278i)T + (41.9 - 9.65i)T^{2} \) |
| 47 | \( 1 + (9.65 + 5.75i)T + (22.3 + 41.3i)T^{2} \) |
| 53 | \( 1 + (-0.0627 + 0.0262i)T + (37.2 - 37.7i)T^{2} \) |
| 59 | \( 1 + (9.22 - 6.23i)T + (21.9 - 54.7i)T^{2} \) |
| 61 | \( 1 + (-2.24 + 0.653i)T + (51.4 - 32.7i)T^{2} \) |
| 67 | \( 1 + (0.251 + 0.0809i)T + (54.4 + 39.0i)T^{2} \) |
| 71 | \( 1 + (0.848 + 11.9i)T + (-70.2 + 10.0i)T^{2} \) |
| 73 | \( 1 + (-1.52 - 0.0431i)T + (72.8 + 4.12i)T^{2} \) |
| 79 | \( 1 + (-1.31 + 1.67i)T + (-18.8 - 76.7i)T^{2} \) |
| 83 | \( 1 + (2.87 + 0.327i)T + (80.8 + 18.6i)T^{2} \) |
| 89 | \( 1 + (-2.19 - 3.34i)T + (-35.5 + 81.6i)T^{2} \) |
| 97 | \( 1 + (-19.6 + 1.11i)T + (96.3 - 10.9i)T^{2} \) |
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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
−11.19863712257704938396614100113, −10.27510899354568265170146412815, −8.787749554005828132248542049606, −7.78980171000797048996149858183, −7.42325576744163023546731301420, −6.34528684381224375317958215798, −5.29157121062982166375043774967, −4.70774024131619306466949440082, −3.44482601734654866335603106947, −2.99626685707593954345458162148,
1.17963671681839982565255634100, 2.20471667661038284320655608598, 3.46409979712362132318490694887, 4.59026719091839027819673903211, 5.10583793904433268400941813099, 6.09779326980610322335821655427, 7.57432976081296959554063100797, 8.224802949278341847787559777396, 9.592528859812595583845652819887, 10.32675024321934780490094812634
