| L(s) = 1 | + (0.248 − 1.39i)2-s + (0.540 + 0.841i)3-s + (−1.87 − 0.692i)4-s + (−2.73 + 1.24i)5-s + (1.30 − 0.543i)6-s + (3.05 − 0.896i)7-s + (−1.43 + 2.44i)8-s + (−0.415 + 0.909i)9-s + (1.05 + 4.11i)10-s + (2.08 − 1.80i)11-s + (−0.432 − 1.95i)12-s + (1.60 − 5.46i)13-s + (−0.488 − 4.47i)14-s + (−2.52 − 1.62i)15-s + (3.04 + 2.59i)16-s + (−0.527 + 3.66i)17-s + ⋯ |
| L(s) = 1 | + (0.175 − 0.984i)2-s + (0.312 + 0.485i)3-s + (−0.938 − 0.346i)4-s + (−1.22 + 0.557i)5-s + (0.533 − 0.221i)6-s + (1.15 − 0.338i)7-s + (−0.505 + 0.862i)8-s + (−0.138 + 0.303i)9-s + (0.334 + 1.30i)10-s + (0.628 − 0.544i)11-s + (−0.124 − 0.563i)12-s + (0.444 − 1.51i)13-s + (−0.130 − 1.19i)14-s + (−0.652 − 0.419i)15-s + (0.760 + 0.649i)16-s + (−0.127 + 0.888i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.467 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.467 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30312 - 0.785358i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30312 - 0.785358i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.248 + 1.39i)T \) |
| 3 | \( 1 + (-0.540 - 0.841i)T \) |
| 23 | \( 1 + (-4.30 + 2.11i)T \) |
| good | 5 | \( 1 + (2.73 - 1.24i)T + (3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (-3.05 + 0.896i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-2.08 + 1.80i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.60 + 5.46i)T + (-10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (0.527 - 3.66i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-7.64 + 1.09i)T + (18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-6.93 - 0.997i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (3.38 + 2.17i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (6.35 + 2.90i)T + (24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (0.0544 + 0.119i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-0.553 - 0.860i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 2.86T + 47T^{2} \) |
| 53 | \( 1 + (-0.242 - 0.826i)T + (-44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (2.84 - 9.68i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-2.26 + 3.52i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-6.73 - 5.83i)T + (9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (2.53 - 2.92i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.359 - 2.49i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (10.8 + 3.19i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (9.55 + 4.36i)T + (54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-12.0 + 7.72i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-3.03 - 6.64i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−10.86112439457767403769001637579, −10.11747071579553022124562683526, −8.770266605272427110009797605590, −8.225872013536238542090292151675, −7.35962823434211424711747197901, −5.63086201590118738880892051263, −4.63921488970514913244901332332, −3.64386003512554054973099566774, −3.03670047495545032239262400291, −1.06742138078501800642592218407,
1.31992461005509346411381745775, 3.47534598431266198217967269416, 4.55517749924547563293967378855, 5.17274985601439196493739314444, 6.75344369715140982603280462807, 7.34083408571967747119072800473, 8.163966813299195502072936020881, 8.853930304050711392037332493294, 9.528793848919006058294937737731, 11.50512904982896116434398055390
