L(s) = 1 | + (−0.196 − 1.40i)2-s + (−0.0178 − 0.181i)3-s + (−1.92 + 0.551i)4-s + (1.42 − 2.66i)5-s + (−0.250 + 0.0608i)6-s + (−2.40 + 1.09i)7-s + (1.15 + 2.58i)8-s + (2.90 − 0.578i)9-s + (−4.01 − 1.47i)10-s + (2.82 − 2.31i)11-s + (0.134 + 0.339i)12-s + (1.74 − 0.930i)13-s + (2.00 + 3.15i)14-s + (−0.510 − 0.211i)15-s + (3.39 − 2.12i)16-s + (6.82 − 2.82i)17-s + ⋯ |
L(s) = 1 | + (−0.139 − 0.990i)2-s + (−0.0103 − 0.104i)3-s + (−0.961 + 0.275i)4-s + (0.637 − 1.19i)5-s + (−0.102 + 0.0248i)6-s + (−0.910 + 0.413i)7-s + (0.407 + 0.913i)8-s + (0.969 − 0.192i)9-s + (−1.27 − 0.465i)10-s + (0.850 − 0.697i)11-s + (0.0388 + 0.0979i)12-s + (0.482 − 0.257i)13-s + (0.536 + 0.843i)14-s + (−0.131 − 0.0545i)15-s + (0.847 − 0.530i)16-s + (1.65 − 0.685i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 + 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.784 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.487272 - 1.40373i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.487272 - 1.40373i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.196 + 1.40i)T \) |
| 7 | \( 1 + (2.40 - 1.09i)T \) |
good | 3 | \( 1 + (0.0178 + 0.181i)T + (-2.94 + 0.585i)T^{2} \) |
| 5 | \( 1 + (-1.42 + 2.66i)T + (-2.77 - 4.15i)T^{2} \) |
| 11 | \( 1 + (-2.82 + 2.31i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.74 + 0.930i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-6.82 + 2.82i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.999 - 3.29i)T + (-15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (1.40 - 2.10i)T + (-8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (7.43 + 6.10i)T + (5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (3.15 + 3.15i)T + 31iT^{2} \) |
| 37 | \( 1 + (6.75 - 2.05i)T + (30.7 - 20.5i)T^{2} \) |
| 41 | \( 1 + (-6.27 - 4.19i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (8.66 + 0.853i)T + (42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (-11.1 + 4.61i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-1.87 + 1.53i)T + (10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (-0.516 + 0.965i)T + (-32.7 - 49.0i)T^{2} \) |
| 61 | \( 1 + (-0.898 - 9.12i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (1.26 + 12.8i)T + (-65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (1.79 - 9.00i)T + (-65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (12.2 - 2.43i)T + (67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-0.855 + 2.06i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (1.66 + 0.506i)T + (69.0 + 46.1i)T^{2} \) |
| 89 | \( 1 + (-1.57 - 2.35i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-6.70 + 6.70i)T - 97iT^{2} \) |
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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
−9.689759905073081378967097551868, −9.273271242904643748235467585185, −8.463299578301814211649456628174, −7.44069976338195227393031185731, −5.92190064353851417951123826832, −5.47098526184918833480249517196, −4.06805676336525367236213401946, −3.38427449497564053859751275612, −1.79817889140821142782726096843, −0.854911280244526829094453710613,
1.56897021004872305585706966811, 3.41434670279861124794502579432, 4.13074680545015948037135723130, 5.51517342721503085317979705691, 6.36929906028665986783359459508, 7.06388033970040058596325925884, 7.41850432317648881060433194347, 8.930511117780925823385088844458, 9.584080973494572061409938259682, 10.34115288042617251286159670792