L(s) = 1 | + (7.25 − 3.37i)2-s + (14.8 + 14.8i)3-s + (41.2 − 48.9i)4-s + (41.0 + 41.0i)5-s + (158. + 57.7i)6-s − 67.2·7-s + (133. − 494. i)8-s − 285. i·9-s + (435. + 159. i)10-s + (−959. + 959. i)11-s + (1.34e3 − 115. i)12-s + (−1.15e3 + 1.15e3i)13-s + (−487. + 226. i)14-s + 1.22e3i·15-s + (−696. − 4.03e3i)16-s − 8.74e3·17-s + ⋯ |
L(s) = 1 | + (0.906 − 0.421i)2-s + (0.551 + 0.551i)3-s + (0.644 − 0.764i)4-s + (0.328 + 0.328i)5-s + (0.732 + 0.267i)6-s − 0.196·7-s + (0.261 − 0.965i)8-s − 0.391i·9-s + (0.435 + 0.159i)10-s + (−0.720 + 0.720i)11-s + (0.777 − 0.0665i)12-s + (−0.524 + 0.524i)13-s + (−0.177 + 0.0827i)14-s + 0.362i·15-s + (−0.169 − 0.985i)16-s − 1.78·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(2.59444 - 0.289005i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.59444 - 0.289005i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-7.25 + 3.37i)T \) |
good | 3 | \( 1 + (-14.8 - 14.8i)T + 729iT^{2} \) |
| 5 | \( 1 + (-41.0 - 41.0i)T + 1.56e4iT^{2} \) |
| 7 | \( 1 + 67.2T + 1.17e5T^{2} \) |
| 11 | \( 1 + (959. - 959. i)T - 1.77e6iT^{2} \) |
| 13 | \( 1 + (1.15e3 - 1.15e3i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 + 8.74e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + (-2.78e3 - 2.78e3i)T + 4.70e7iT^{2} \) |
| 23 | \( 1 - 1.80e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + (-1.49e4 + 1.49e4i)T - 5.94e8iT^{2} \) |
| 31 | \( 1 - 3.31e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + (4.36e3 + 4.36e3i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 - 2.35e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (-6.16e4 + 6.16e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 - 1.35e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + (-1.73e5 - 1.73e5i)T + 2.21e10iT^{2} \) |
| 59 | \( 1 + (4.91e4 - 4.91e4i)T - 4.21e10iT^{2} \) |
| 61 | \( 1 + (-1.90e5 + 1.90e5i)T - 5.15e10iT^{2} \) |
| 67 | \( 1 + (2.46e5 + 2.46e5i)T + 9.04e10iT^{2} \) |
| 71 | \( 1 + 5.72e3T + 1.28e11T^{2} \) |
| 73 | \( 1 + 3.44e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 7.85e4iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (-4.72e5 - 4.72e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 - 4.18e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 7.58e5T + 8.32e11T^{2} \) |
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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
−17.86788926103715178953618445683, −15.82102094913555157545593261065, −14.93473941151050788312807140948, −13.76367784467897541111718653002, −12.36945401281111959257007079956, −10.65490174670902324785894097916, −9.387601398460391390117834471618, −6.71993881450919304869570856618, −4.56020044533902974775716321148, −2.64601104547139974863311474580,
2.65062918024158642333562218180, 5.17114502792778189696594804040, 7.10913357216116933293892147752, 8.598775435386475422089012529790, 11.08365464521690195687279468932, 13.08479693603125211564324792459, 13.39697825708219416443469857381, 15.00352801814481136728722638522, 16.27734066736252497737106511946, 17.64214258361500712653603491637