| L(s) = 1 | + (0.0125 − 0.0872i)2-s + (0.415 + 0.909i)3-s + (1.91 + 0.561i)4-s + (0.0845 − 0.0248i)6-s + (1.30 + 0.835i)7-s + (0.146 − 0.320i)8-s + (1.30 − 1.51i)9-s + (0.289 + 2.01i)11-s + (0.283 + 1.97i)12-s + (−1.80 + 1.15i)13-s + (0.0892 − 0.103i)14-s + (3.32 + 2.13i)16-s + (4.18 − 1.22i)17-s + (−0.115 − 0.133i)18-s + (−7.12 − 2.09i)19-s + ⋯ |
| L(s) = 1 | + (0.00887 − 0.0617i)2-s + (0.239 + 0.525i)3-s + (0.955 + 0.280i)4-s + (0.0345 − 0.0101i)6-s + (0.491 + 0.315i)7-s + (0.0516 − 0.113i)8-s + (0.436 − 0.503i)9-s + (0.0872 + 0.607i)11-s + (0.0818 + 0.569i)12-s + (−0.499 + 0.320i)13-s + (0.0238 − 0.0275i)14-s + (0.831 + 0.534i)16-s + (1.01 − 0.297i)17-s + (−0.0272 − 0.0314i)18-s + (−1.63 − 0.479i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.97610 + 0.721661i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.97610 + 0.721661i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 23 | \( 1 + (-2.79 + 3.89i)T \) |
| good | 2 | \( 1 + (-0.0125 + 0.0872i)T + (-1.91 - 0.563i)T^{2} \) |
| 3 | \( 1 + (-0.415 - 0.909i)T + (-1.96 + 2.26i)T^{2} \) |
| 7 | \( 1 + (-1.30 - 0.835i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.289 - 2.01i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (1.80 - 1.15i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-4.18 + 1.22i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (7.12 + 2.09i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (4.30 - 1.26i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (0.376 - 0.824i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (7.26 - 8.38i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-3.95 - 4.56i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (2.24 + 4.90i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 2.72T + 47T^{2} \) |
| 53 | \( 1 + (-0.230 - 0.148i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-6.74 + 4.33i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (1.33 - 2.92i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (0.279 - 1.94i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.58 + 11.0i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (9.95 + 2.92i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-2.70 + 1.74i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (2.85 - 3.29i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-0.266 - 0.584i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (3.62 + 4.18i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.70964646902315213920962950410, −10.09563524425232181095768396940, −9.092755897488418591735286153618, −8.217651138529913887114228469979, −7.11851087935887285611836262002, −6.53052538645481493524864409059, −5.12496970965445915451372079700, −4.13037553257989595136588621195, −2.94634123839830705337910499163, −1.77179794535964891101190759377,
1.39484106495637282561370142728, 2.40987188559336149600019140500, 3.81229096337852668977692793695, 5.26389662012439240367404443205, 6.12190906872972244060322772843, 7.28119667572638627930939917532, 7.68225021502971027013223361682, 8.644419086889605294070106305548, 10.01772381339057714708279032377, 10.69008088539092977675309631476
