| L(s) = 1 | + (2.26 − 0.118i)2-s + (0.104 − 0.994i)3-s + (3.11 − 0.327i)4-s + (−1.01 − 1.98i)5-s + (0.118 − 2.26i)6-s + (−1.72 − 1.39i)7-s + (2.53 − 0.400i)8-s + (−0.978 − 0.207i)9-s + (−2.51 − 4.36i)10-s + (3.29 + 0.396i)11-s − 3.13i·12-s + (3.48 + 0.940i)13-s + (−4.05 − 2.94i)14-s + (−2.07 + 0.797i)15-s + (−0.449 + 0.0955i)16-s + (1.66 + 1.84i)17-s + ⋯ |
| L(s) = 1 | + (1.59 − 0.0838i)2-s + (0.0603 − 0.574i)3-s + (1.55 − 0.163i)4-s + (−0.451 − 0.886i)5-s + (0.0483 − 0.923i)6-s + (−0.650 − 0.526i)7-s + (0.894 − 0.141i)8-s + (−0.326 − 0.0693i)9-s + (−0.796 − 1.38i)10-s + (0.992 + 0.119i)11-s − 0.903i·12-s + (0.965 + 0.260i)13-s + (−1.08 − 0.788i)14-s + (−0.536 + 0.205i)15-s + (−0.112 + 0.0238i)16-s + (0.403 + 0.447i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.44460 - 1.71831i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.44460 - 1.71831i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.104 + 0.994i)T \) |
| 11 | \( 1 + (-3.29 - 0.396i)T \) |
| 13 | \( 1 + (-3.48 - 0.940i)T \) |
| good | 2 | \( 1 + (-2.26 + 0.118i)T + (1.98 - 0.209i)T^{2} \) |
| 5 | \( 1 + (1.01 + 1.98i)T + (-2.93 + 4.04i)T^{2} \) |
| 7 | \( 1 + (1.72 + 1.39i)T + (1.45 + 6.84i)T^{2} \) |
| 17 | \( 1 + (-1.66 - 1.84i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (0.665 + 0.255i)T + (14.1 + 12.7i)T^{2} \) |
| 23 | \( 1 + (1.06 - 0.613i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.60 - 3.61i)T + (-19.4 - 21.5i)T^{2} \) |
| 31 | \( 1 + (-7.14 - 3.63i)T + (18.2 + 25.0i)T^{2} \) |
| 37 | \( 1 + (-2.21 - 5.76i)T + (-27.4 + 24.7i)T^{2} \) |
| 41 | \( 1 + (6.14 - 4.97i)T + (8.52 - 40.1i)T^{2} \) |
| 43 | \( 1 + (1.74 - 3.02i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.669 + 4.22i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (-0.774 - 2.38i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-5.74 + 7.09i)T + (-12.2 - 57.7i)T^{2} \) |
| 61 | \( 1 + (-5.39 + 4.86i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (7.50 + 2.00i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (11.5 + 0.607i)T + (70.6 + 7.42i)T^{2} \) |
| 73 | \( 1 + (-0.0602 + 0.380i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (1.58 - 0.514i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-6.16 + 3.14i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (-2.57 + 9.61i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-2.91 + 1.89i)T + (39.4 - 88.6i)T^{2} \) |
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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
−11.68692239864465332684481142848, −10.35565624614181111158943274619, −9.021096129939872476643204644924, −8.138541477769086324219471325530, −6.71222814424362448653054400393, −6.31901382510101365557141171417, −5.00595378259181686657251369767, −4.04053279119457433874313701226, −3.25959660016554439862091396958, −1.38999179030859905939100677749,
2.70252636894768825483514716015, 3.54650350439421041617815095453, 4.24384870390482022434511327951, 5.66967576203555231219117255298, 6.27621521600661654274993053759, 7.20388637568908249816175921784, 8.627684702935368110081719653239, 9.680332394263085492280225152621, 10.79128047802517769833808586189, 11.60691643411169487769980934928
