| L(s) = 1 | + (0.766 − 0.642i)2-s + (0.247 + 1.71i)3-s + (0.173 − 0.984i)4-s + (−1.96 + 0.714i)5-s + (1.29 + 1.15i)6-s + (−0.696 − 3.95i)7-s + (−0.500 − 0.866i)8-s + (−2.87 + 0.848i)9-s + (−1.04 + 1.80i)10-s + (−0.199 − 0.0726i)11-s + (1.73 + 0.0539i)12-s + (3.98 + 3.34i)13-s + (−3.07 − 2.57i)14-s + (−1.70 − 3.18i)15-s + (−0.939 − 0.342i)16-s + (−1.89 + 3.27i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (0.142 + 0.989i)3-s + (0.0868 − 0.492i)4-s + (−0.877 + 0.319i)5-s + (0.527 + 0.471i)6-s + (−0.263 − 1.49i)7-s + (−0.176 − 0.306i)8-s + (−0.959 + 0.282i)9-s + (−0.330 + 0.571i)10-s + (−0.0601 − 0.0219i)11-s + (0.499 + 0.0155i)12-s + (1.10 + 0.926i)13-s + (−0.821 − 0.689i)14-s + (−0.441 − 0.822i)15-s + (−0.234 − 0.0855i)16-s + (−0.459 + 0.795i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.995702 - 0.0175504i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.995702 - 0.0175504i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 3 | \( 1 + (-0.247 - 1.71i)T \) |
| good | 5 | \( 1 + (1.96 - 0.714i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.696 + 3.95i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (0.199 + 0.0726i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-3.98 - 3.34i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.89 - 3.27i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.636 - 1.10i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.144 + 0.816i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-7.05 + 5.92i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.614 + 3.48i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (1.77 - 3.07i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.09 - 1.76i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (6.48 + 2.35i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.221 + 1.25i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + (-8.04 + 2.92i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.492 - 2.79i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-7.47 - 6.27i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-5.86 + 10.1i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.375 + 0.649i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.81 - 1.52i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (8.73 - 7.32i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (2.69 + 4.67i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.8 - 4.29i)T + (74.3 + 62.3i)T^{2} \) |
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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
−15.37401773073748770596956734883, −14.17927167451174065635525086233, −13.38521833081281826128298901048, −11.60690598464516900598705332279, −10.85974638153197744956557063727, −9.893503000809830783045294093469, −8.217318357166537854602715112056, −6.50108602103066189607195962178, −4.33265229555500093110670406066, −3.63529219061052785628640177139,
3.04501983622561113941906123750, 5.34023075222685362304283042385, 6.62127173824416778554697015169, 8.102005943210790720610925348194, 8.842157567154690045122610516880, 11.36155494229811508115626261555, 12.26821563829429255355728167909, 12.98476108101894717368158986072, 14.23249055103669590544808526037, 15.58396598576397982519707717615
