L(s) = 1 | + (−0.365 − 0.563i)2-s + (−0.784 − 2.04i)3-s + (0.629 − 1.41i)4-s + (1.68 − 1.46i)5-s + (−0.864 + 1.19i)6-s + (0.965 + 2.46i)7-s + (−2.35 + 0.372i)8-s + (−1.33 + 1.20i)9-s + (−1.44 − 0.414i)10-s + (1.59 − 1.76i)11-s + (−3.38 − 0.177i)12-s + (−4.81 + 2.45i)13-s + (1.03 − 1.44i)14-s + (−4.32 − 2.30i)15-s + (−1.00 − 1.11i)16-s + (3.46 + 4.27i)17-s + ⋯ |
L(s) = 1 | + (−0.258 − 0.398i)2-s + (−0.453 − 1.18i)3-s + (0.314 − 0.707i)4-s + (0.754 − 0.655i)5-s + (−0.353 + 0.486i)6-s + (0.364 + 0.931i)7-s + (−0.832 + 0.131i)8-s + (−0.445 + 0.400i)9-s + (−0.456 − 0.131i)10-s + (0.480 − 0.533i)11-s + (−0.977 − 0.0512i)12-s + (−1.33 + 0.680i)13-s + (0.276 − 0.386i)14-s + (−1.11 − 0.594i)15-s + (−0.250 − 0.277i)16-s + (0.839 + 1.03i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 + 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.619 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.457796 - 0.944979i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.457796 - 0.944979i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.68 + 1.46i)T \) |
| 7 | \( 1 + (-0.965 - 2.46i)T \) |
good | 2 | \( 1 + (0.365 + 0.563i)T + (-0.813 + 1.82i)T^{2} \) |
| 3 | \( 1 + (0.784 + 2.04i)T + (-2.22 + 2.00i)T^{2} \) |
| 11 | \( 1 + (-1.59 + 1.76i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (4.81 - 2.45i)T + (7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-3.46 - 4.27i)T + (-3.53 + 16.6i)T^{2} \) |
| 19 | \( 1 + (0.578 - 0.257i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-3.37 + 2.18i)T + (9.35 - 21.0i)T^{2} \) |
| 29 | \( 1 + (-3.54 - 4.87i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (0.869 + 0.0913i)T + (30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (0.210 - 4.01i)T + (-36.7 - 3.86i)T^{2} \) |
| 41 | \( 1 + (-4.20 - 1.36i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-4.81 + 4.81i)T - 43iT^{2} \) |
| 47 | \( 1 + (7.66 + 6.20i)T + (9.77 + 45.9i)T^{2} \) |
| 53 | \( 1 + (-9.74 + 3.74i)T + (39.3 - 35.4i)T^{2} \) |
| 59 | \( 1 + (-3.31 + 0.705i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (0.387 - 1.82i)T + (-55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (-7.37 + 5.97i)T + (13.9 - 65.5i)T^{2} \) |
| 71 | \( 1 + (5.22 - 3.79i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (9.02 - 0.473i)T + (72.6 - 7.63i)T^{2} \) |
| 79 | \( 1 + (10.2 - 1.07i)T + (77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (1.39 + 8.79i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-14.2 - 3.01i)T + (81.3 + 36.1i)T^{2} \) |
| 97 | \( 1 + (1.14 - 7.25i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−12.16911419619043794549725269291, −11.73569256948949915818744582776, −10.39454787677943584451606076314, −9.355501509782712646275423348196, −8.450464478300125982578486247120, −6.86525164219438372645724635109, −5.99648520096723783550548014629, −5.13886349184871448066480614721, −2.31017375963976638356778960570, −1.27838138331441544521075734734,
2.88819935372845018302440638261, 4.32972942323316472536725606261, 5.53138963106152622464116073392, 7.02144383651447117736262903342, 7.64282367695904238768563333220, 9.430303338329979460890500082146, 9.960795317173659924355192130960, 10.92454811963538884142337448552, 11.83957162595419650024241546225, 13.05389445082625058901206511155