| 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
−13.05389445082625058901206511155, −11.83957162595419650024241546225, −10.92454811963538884142337448552, −9.960795317173659924355192130960, −9.430303338329979460890500082146, −7.64282367695904238768563333220, −7.02144383651447117736262903342, −5.53138963106152622464116073392, −4.32972942323316472536725606261, −2.88819935372845018302440638261,
1.27838138331441544521075734734, 2.31017375963976638356778960570, 5.13886349184871448066480614721, 5.99648520096723783550548014629, 6.86525164219438372645724635109, 8.450464478300125982578486247120, 9.355501509782712646275423348196, 10.39454787677943584451606076314, 11.73569256948949915818744582776, 12.16911419619043794549725269291
