| L(s) = 1 | + (1.27 − 2.00i)2-s + (−1.02 + 0.962i)3-s + (−1.55 − 3.29i)4-s + (0.193 − 2.22i)5-s + (0.626 + 3.28i)6-s + (1.19 − 0.869i)7-s + (−3.87 − 0.489i)8-s + (−0.0641 + 1.01i)9-s + (−4.22 − 3.22i)10-s + (−1.09 + 1.72i)11-s + (4.76 + 1.88i)12-s + (−0.0311 + 0.495i)13-s + (−0.220 − 3.50i)14-s + (1.94 + 2.47i)15-s + (−1.27 + 1.53i)16-s + (−0.275 + 0.585i)17-s + ⋯ |
| L(s) = 1 | + (0.900 − 1.41i)2-s + (−0.592 + 0.555i)3-s + (−0.776 − 1.64i)4-s + (0.0865 − 0.996i)5-s + (0.255 + 1.34i)6-s + (0.452 − 0.328i)7-s + (−1.37 − 0.173i)8-s + (−0.0213 + 0.339i)9-s + (−1.33 − 1.01i)10-s + (−0.330 + 0.520i)11-s + (1.37 + 0.544i)12-s + (−0.00864 + 0.137i)13-s + (−0.0589 − 0.937i)14-s + (0.502 + 0.637i)15-s + (−0.318 + 0.384i)16-s + (−0.0667 + 0.141i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.816277 - 1.11901i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.816277 - 1.11901i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.193 + 2.22i)T \) |
| good | 2 | \( 1 + (-1.27 + 2.00i)T + (-0.851 - 1.80i)T^{2} \) |
| 3 | \( 1 + (1.02 - 0.962i)T + (0.188 - 2.99i)T^{2} \) |
| 7 | \( 1 + (-1.19 + 0.869i)T + (2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (1.09 - 1.72i)T + (-4.68 - 9.95i)T^{2} \) |
| 13 | \( 1 + (0.0311 - 0.495i)T + (-12.8 - 1.62i)T^{2} \) |
| 17 | \( 1 + (0.275 - 0.585i)T + (-10.8 - 13.0i)T^{2} \) |
| 19 | \( 1 + (-6.12 - 5.75i)T + (1.19 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-4.49 - 1.15i)T + (20.1 + 11.0i)T^{2} \) |
| 29 | \( 1 + (5.77 + 3.17i)T + (15.5 + 24.4i)T^{2} \) |
| 31 | \( 1 + (3.50 - 7.43i)T + (-19.7 - 23.8i)T^{2} \) |
| 37 | \( 1 + (-3.40 + 4.11i)T + (-6.93 - 36.3i)T^{2} \) |
| 41 | \( 1 + (8.24 - 2.11i)T + (35.9 - 19.7i)T^{2} \) |
| 43 | \( 1 + (-0.692 + 2.13i)T + (-34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (3.77 - 0.476i)T + (45.5 - 11.6i)T^{2} \) |
| 53 | \( 1 + (-2.60 + 13.6i)T + (-49.2 - 19.5i)T^{2} \) |
| 59 | \( 1 + (5.50 + 2.17i)T + (43.0 + 40.3i)T^{2} \) |
| 61 | \( 1 + (-3.60 - 0.925i)T + (53.4 + 29.3i)T^{2} \) |
| 67 | \( 1 + (7.73 - 4.25i)T + (35.9 - 56.5i)T^{2} \) |
| 71 | \( 1 + (-11.6 + 1.46i)T + (68.7 - 17.6i)T^{2} \) |
| 73 | \( 1 + (7.46 - 2.95i)T + (53.2 - 49.9i)T^{2} \) |
| 79 | \( 1 + (-1.47 + 1.38i)T + (4.96 - 78.8i)T^{2} \) |
| 83 | \( 1 + (-10.7 - 10.1i)T + (5.21 + 82.8i)T^{2} \) |
| 89 | \( 1 + (-12.3 + 4.89i)T + (64.8 - 60.9i)T^{2} \) |
| 97 | \( 1 + (1.15 + 0.633i)T + (51.9 + 81.8i)T^{2} \) |
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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
−12.92094648513313087901365241231, −11.98464734418224120687312015816, −11.23459832803683186451066522686, −10.27722350574827011332059930660, −9.471506944819161512562272602278, −7.79934327638678890253351020603, −5.43235157549015221329966305754, −4.94361896098094216891991019759, −3.76113729350178289810944269258, −1.66520141589999034517893803833,
3.26400172356835298575951542910, 5.11150985607247848612828900640, 5.99415997221431360640247058660, 6.97556719518386496930027803824, 7.66220632538985130042390858950, 9.178466660098613897276402918608, 11.01385641861982328370966826624, 11.80450903038255503556242183322, 13.12363104894184849591624960367, 13.72361301387009220377557184527
