| L(s) = 1 | + (−0.0571 + 1.09i)2-s + (−2.14 + 1.73i)3-s + (0.801 + 0.0842i)4-s + (−1.85 − 1.24i)5-s + (−1.76 − 2.43i)6-s + (−2.33 + 1.24i)7-s + (−0.479 + 3.02i)8-s + (0.954 − 4.48i)9-s + (1.46 − 1.95i)10-s + (−2.33 + 0.496i)11-s + (−1.86 + 1.20i)12-s + (1.87 − 3.67i)13-s + (−1.22 − 2.61i)14-s + (6.13 − 0.545i)15-s + (−1.70 − 0.361i)16-s + (−4.38 + 1.68i)17-s + ⋯ |
| L(s) = 1 | + (−0.0404 + 0.771i)2-s + (−1.23 + 1.00i)3-s + (0.400 + 0.0421i)4-s + (−0.829 − 0.557i)5-s + (−0.722 − 0.994i)6-s + (−0.882 + 0.470i)7-s + (−0.169 + 1.07i)8-s + (0.318 − 1.49i)9-s + (0.464 − 0.617i)10-s + (−0.703 + 0.149i)11-s + (−0.537 + 0.349i)12-s + (0.518 − 1.01i)13-s + (−0.327 − 0.699i)14-s + (1.58 − 0.140i)15-s + (−0.425 − 0.0903i)16-s + (−1.06 + 0.407i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 + 0.459i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 + 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0982173 - 0.403526i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0982173 - 0.403526i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.85 + 1.24i)T \) |
| 7 | \( 1 + (2.33 - 1.24i)T \) |
| good | 2 | \( 1 + (0.0571 - 1.09i)T + (-1.98 - 0.209i)T^{2} \) |
| 3 | \( 1 + (2.14 - 1.73i)T + (0.623 - 2.93i)T^{2} \) |
| 11 | \( 1 + (2.33 - 0.496i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-1.87 + 3.67i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (4.38 - 1.68i)T + (12.6 - 11.3i)T^{2} \) |
| 19 | \( 1 + (-0.387 - 3.68i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-3.37 - 0.176i)T + (22.8 + 2.40i)T^{2} \) |
| 29 | \( 1 + (4.02 - 5.54i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (3.81 - 8.56i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-1.82 - 2.80i)T + (-15.0 + 33.8i)T^{2} \) |
| 41 | \( 1 + (-4.39 + 1.42i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-3.92 + 3.92i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.93 - 7.63i)T + (-34.9 - 31.4i)T^{2} \) |
| 53 | \( 1 + (3.29 + 4.06i)T + (-11.0 + 51.8i)T^{2} \) |
| 59 | \( 1 + (-3.33 + 3.70i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-2.25 + 2.03i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (4.08 + 10.6i)T + (-49.7 + 44.8i)T^{2} \) |
| 71 | \( 1 + (2.61 + 1.90i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-12.8 - 8.31i)T + (29.6 + 66.6i)T^{2} \) |
| 79 | \( 1 + (-0.754 - 1.69i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-4.83 - 0.765i)T + (78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (2.32 + 2.58i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (3.70 - 0.586i)T + (92.2 - 29.9i)T^{2} \) |
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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
−12.87589450299509813211134946541, −12.29405243040200108657014622101, −11.06191878840843384195936115172, −10.65976270689199704754872567788, −9.202414738241946538357483792680, −8.115707903086363583783998895266, −6.80995754270820579648886892789, −5.72526107573897561843174602107, −5.03083547434985282559900670783, −3.44448395747090503392528282363,
0.42178289772861220174247427385, 2.49316241634651329586149825606, 4.11409978697395123688638795791, 6.07867708680543518017610470438, 6.87524117547548450620725025785, 7.47291505677687849216028099689, 9.427454073454106832812164951021, 10.86396355907463046165956235620, 11.17815228163579700543236484158, 11.85521432194875878739468603608
