| L(s) = 1 | + (0.0705 + 0.400i)2-s + (−0.765 + 0.536i)3-s + (1.72 − 0.627i)4-s + (0.310 − 2.21i)5-s + (−0.268 − 0.268i)6-s + (−0.428 − 4.90i)7-s + (0.779 + 1.34i)8-s + (−0.727 + 1.99i)9-s + (0.907 − 0.0320i)10-s + (0.406 − 0.234i)11-s + (−0.984 + 1.40i)12-s + (0.996 − 0.362i)13-s + (1.93 − 0.517i)14-s + (0.949 + 1.86i)15-s + (2.32 − 1.95i)16-s + (−1.18 + 3.25i)17-s + ⋯ |
| L(s) = 1 | + (0.0498 + 0.282i)2-s + (−0.442 + 0.309i)3-s + (0.862 − 0.313i)4-s + (0.138 − 0.990i)5-s + (−0.109 − 0.109i)6-s + (−0.162 − 1.85i)7-s + (0.275 + 0.477i)8-s + (−0.242 + 0.665i)9-s + (0.287 − 0.0101i)10-s + (0.122 − 0.0708i)11-s + (−0.284 + 0.405i)12-s + (0.276 − 0.100i)13-s + (0.516 − 0.138i)14-s + (0.245 + 0.480i)15-s + (0.581 − 0.488i)16-s + (−0.287 + 0.789i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22941 - 0.269918i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22941 - 0.269918i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.310 + 2.21i)T \) |
| 37 | \( 1 + (5.98 - 1.07i)T \) |
| good | 2 | \( 1 + (-0.0705 - 0.400i)T + (-1.87 + 0.684i)T^{2} \) |
| 3 | \( 1 + (0.765 - 0.536i)T + (1.02 - 2.81i)T^{2} \) |
| 7 | \( 1 + (0.428 + 4.90i)T + (-6.89 + 1.21i)T^{2} \) |
| 11 | \( 1 + (-0.406 + 0.234i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.996 + 0.362i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (1.18 - 3.25i)T + (-13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (-3.78 - 5.40i)T + (-6.49 + 17.8i)T^{2} \) |
| 23 | \( 1 + (0.298 - 0.516i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.09 - 4.09i)T + (-25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (-1.70 + 1.70i)T - 31iT^{2} \) |
| 41 | \( 1 + (-0.559 - 1.53i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + 2.50T + 43T^{2} \) |
| 47 | \( 1 + (-6.32 + 1.69i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.19 - 13.6i)T + (-52.1 - 9.20i)T^{2} \) |
| 59 | \( 1 + (-0.467 + 5.33i)T + (-58.1 - 10.2i)T^{2} \) |
| 61 | \( 1 + (-4.91 + 10.5i)T + (-39.2 - 46.7i)T^{2} \) |
| 67 | \( 1 + (-4.60 + 0.403i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (1.87 - 10.6i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (1.78 + 1.78i)T + 73iT^{2} \) |
| 79 | \( 1 + (-3.11 + 0.272i)T + (77.7 - 13.7i)T^{2} \) |
| 83 | \( 1 + (4.78 - 2.23i)T + (53.3 - 63.5i)T^{2} \) |
| 89 | \( 1 + (4.57 + 0.400i)T + (87.6 + 15.4i)T^{2} \) |
| 97 | \( 1 + (-9.02 - 5.20i)T + (48.5 + 84.0i)T^{2} \) |
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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
−12.52920156285870664360247970917, −11.36112435431161065302533350724, −10.55617261861432240645028248452, −9.925851332969793828692489775874, −8.221641937253536378718762996701, −7.38936213217784632931139244086, −6.13672952657613490232550710374, −5.10576288544345698521456642903, −3.85060383721851567101666093209, −1.40401971019748552085620725446,
2.33330843559902752508842559392, 3.22442667092290471126233926204, 5.53796418213840370657617976093, 6.44459161184187001330211495087, 7.18440567773152190769979612327, 8.757025282708639912191350235611, 9.749565817321518228688008428859, 11.17203245342950173724657049951, 11.69819678245775546549737686162, 12.24582925107784672153612253889
