| L(s) = 1 | + (1.52 + 1.42i)2-s + (0.980 − 1.42i)3-s + (0.160 + 2.34i)4-s + (0.519 − 1.00i)5-s + (3.52 − 0.781i)6-s + (−4.03 + 1.75i)7-s + (−0.464 + 0.570i)8-s + (−1.07 − 2.79i)9-s + (2.21 − 0.788i)10-s + (−0.344 + 1.65i)11-s + (3.50 + 2.07i)12-s + (−2.42 + 3.97i)13-s + (−8.63 − 3.07i)14-s + (−0.921 − 1.72i)15-s + (3.14 − 0.431i)16-s + (−2.32 + 0.484i)17-s + ⋯ |
| L(s) = 1 | + (1.07 + 1.00i)2-s + (0.565 − 0.824i)3-s + (0.0802 + 1.17i)4-s + (0.232 − 0.448i)5-s + (1.44 − 0.318i)6-s + (−1.52 + 0.661i)7-s + (−0.164 + 0.201i)8-s + (−0.359 − 0.933i)9-s + (0.701 − 0.249i)10-s + (−0.103 + 0.499i)11-s + (1.01 + 0.598i)12-s + (−0.671 + 1.10i)13-s + (−2.30 − 0.820i)14-s + (−0.237 − 0.444i)15-s + (0.785 − 0.107i)16-s + (−0.565 + 0.117i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 - 0.539i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.842 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.84670 + 0.540562i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.84670 + 0.540562i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.980 + 1.42i)T \) |
| 47 | \( 1 + (6.84 - 0.279i)T \) |
| good | 2 | \( 1 + (-1.52 - 1.42i)T + (0.136 + 1.99i)T^{2} \) |
| 5 | \( 1 + (-0.519 + 1.00i)T + (-2.88 - 4.08i)T^{2} \) |
| 7 | \( 1 + (4.03 - 1.75i)T + (4.77 - 5.11i)T^{2} \) |
| 11 | \( 1 + (0.344 - 1.65i)T + (-10.0 - 4.38i)T^{2} \) |
| 13 | \( 1 + (2.42 - 3.97i)T + (-5.98 - 11.5i)T^{2} \) |
| 17 | \( 1 + (2.32 - 0.484i)T + (15.5 - 6.77i)T^{2} \) |
| 19 | \( 1 + (-3.17 + 1.64i)T + (10.9 - 15.5i)T^{2} \) |
| 23 | \( 1 + (2.67 + 2.86i)T + (-1.56 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-4.58 + 2.78i)T + (13.3 - 25.7i)T^{2} \) |
| 31 | \( 1 + (0.828 + 6.03i)T + (-29.8 + 8.36i)T^{2} \) |
| 37 | \( 1 + (-0.0503 - 0.141i)T + (-28.7 + 23.3i)T^{2} \) |
| 41 | \( 1 + (-8.17 + 6.65i)T + (8.34 - 40.1i)T^{2} \) |
| 43 | \( 1 + (-5.86 + 0.401i)T + (42.5 - 5.85i)T^{2} \) |
| 53 | \( 1 + (-0.776 - 0.954i)T + (-10.7 + 51.8i)T^{2} \) |
| 59 | \( 1 + (10.0 + 0.688i)T + (58.4 + 8.03i)T^{2} \) |
| 61 | \( 1 + (2.67 - 7.52i)T + (-47.3 - 38.4i)T^{2} \) |
| 67 | \( 1 + (4.43 - 10.2i)T + (-45.7 - 48.9i)T^{2} \) |
| 71 | \( 1 + (6.78 - 6.33i)T + (4.84 - 70.8i)T^{2} \) |
| 73 | \( 1 + (0.508 + 1.81i)T + (-62.3 + 37.9i)T^{2} \) |
| 79 | \( 1 + (5.25 - 7.43i)T + (-26.4 - 74.4i)T^{2} \) |
| 83 | \( 1 + (-10.9 - 2.26i)T + (76.1 + 33.0i)T^{2} \) |
| 89 | \( 1 + (-4.64 - 2.40i)T + (51.3 + 72.7i)T^{2} \) |
| 97 | \( 1 + (-9.02 - 1.24i)T + (93.4 + 26.1i)T^{2} \) |
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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
−13.29934058591188279237112341833, −12.65172449833329690006473847368, −11.98621811403288300722786174818, −9.712324720340620083579637342508, −8.971518200733742445736641176148, −7.44510862663299361445339849647, −6.63554993580291193553183356210, −5.80316149078865627310010110934, −4.25648431513986820968497347180, −2.64645952252806430287621727678,
2.89273131023658062019813839718, 3.37594890437169160125686246208, 4.76906372279411165373561145506, 6.10963275622272383763804897581, 7.75941605368105920387109718161, 9.440787724172788195694650343957, 10.31704378610237946205100130196, 10.78649689199644566426164900400, 12.25547161619841316432371509487, 13.15651627368556752441544994701
