| L(s) = 1 | + (0.727 + 0.807i)2-s + (0.0855 − 0.813i)4-s + (−0.934 + 2.03i)5-s + (1.73 − 3.01i)7-s + (2.47 − 1.80i)8-s + (−2.32 + 0.722i)10-s + (−2.35 − 2.61i)11-s + (1.45 − 1.61i)13-s + (3.69 − 0.786i)14-s + (1.65 + 0.352i)16-s + (1.54 − 1.12i)17-s + (−0.970 + 0.704i)19-s + (1.57 + 0.934i)20-s + (0.399 − 3.80i)22-s + (1.96 − 0.417i)23-s + ⋯ |
| L(s) = 1 | + (0.514 + 0.571i)2-s + (0.0427 − 0.406i)4-s + (−0.417 + 0.908i)5-s + (0.657 − 1.13i)7-s + (0.876 − 0.636i)8-s + (−0.733 + 0.228i)10-s + (−0.709 − 0.788i)11-s + (0.402 − 0.447i)13-s + (0.988 − 0.210i)14-s + (0.414 + 0.0880i)16-s + (0.375 − 0.272i)17-s + (−0.222 + 0.161i)19-s + (0.351 + 0.208i)20-s + (0.0852 − 0.810i)22-s + (0.409 − 0.0871i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.95956 - 0.328355i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.95956 - 0.328355i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.934 - 2.03i)T \) |
| good | 2 | \( 1 + (-0.727 - 0.807i)T + (-0.209 + 1.98i)T^{2} \) |
| 7 | \( 1 + (-1.73 + 3.01i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.35 + 2.61i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-1.45 + 1.61i)T + (-1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-1.54 + 1.12i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (0.970 - 0.704i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.96 + 0.417i)T + (21.0 - 9.35i)T^{2} \) |
| 29 | \( 1 + (-3.26 + 1.45i)T + (19.4 - 21.5i)T^{2} \) |
| 31 | \( 1 + (-5.79 - 2.58i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-2.46 - 7.58i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.73 - 1.92i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (-5.34 + 9.26i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (11.9 - 5.34i)T + (31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (-11.2 - 8.18i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-3.64 + 4.04i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-0.353 - 0.392i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (9.83 + 4.37i)T + (44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (6.38 + 4.63i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (2.75 - 8.46i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-6.17 + 2.74i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (0.372 + 3.53i)T + (-81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 + (2.71 - 8.36i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (15.6 - 6.98i)T + (64.9 - 72.0i)T^{2} \) |
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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
−10.58892950261605793007370915032, −9.968872661050406280385824630531, −8.309423163609366047050396163891, −7.68238842871734753282639907441, −6.85249991309586083389819845411, −6.05457427816342572638892619338, −4.98969438217513583273946369308, −4.07576768856207177088538654304, −2.94435893272798055123587184455, −0.997917929839179098789967450537,
1.72142100010708419542047706029, 2.75987274592096077730670106628, 4.14639469237554963047466082197, 4.84539124772393490032278838845, 5.65955688267986993573883629132, 7.20580970712811217826717862369, 8.205356932311507885886372726273, 8.572589273646557034433264688781, 9.674311222888204262596403534286, 10.84950831729204971668079357538
