| 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
−10.84950831729204971668079357538, −9.674311222888204262596403534286, −8.572589273646557034433264688781, −8.205356932311507885886372726273, −7.20580970712811217826717862369, −5.65955688267986993573883629132, −4.84539124772393490032278838845, −4.14639469237554963047466082197, −2.75987274592096077730670106628, −1.72142100010708419542047706029,
0.997917929839179098789967450537, 2.94435893272798055123587184455, 4.07576768856207177088538654304, 4.98969438217513583273946369308, 6.05457427816342572638892619338, 6.85249991309586083389819845411, 7.68238842871734753282639907441, 8.309423163609366047050396163891, 9.968872661050406280385824630531, 10.58892950261605793007370915032
