L(s) = 1 | + (2.70 + 0.813i)2-s + (−2.61 + 2.61i)3-s + (6.67 + 4.40i)4-s + (−0.435 − 11.1i)5-s + (−9.22 + 4.96i)6-s + (−17.7 − 17.7i)7-s + (14.4 + 17.3i)8-s + 13.2i·9-s + (7.91 − 30.6i)10-s + 7.37i·11-s + (−29.0 + 5.93i)12-s + (−2.68 − 2.68i)13-s + (−33.6 − 62.6i)14-s + (30.3 + 28.1i)15-s + (25.1 + 58.8i)16-s + (−20.2 + 20.2i)17-s + ⋯ |
L(s) = 1 | + (0.957 + 0.287i)2-s + (−0.503 + 0.503i)3-s + (0.834 + 0.551i)4-s + (−0.0389 − 0.999i)5-s + (−0.627 + 0.337i)6-s + (−0.959 − 0.959i)7-s + (0.640 + 0.767i)8-s + 0.492i·9-s + (0.250 − 0.968i)10-s + 0.202i·11-s + (−0.698 + 0.142i)12-s + (−0.0572 − 0.0572i)13-s + (−0.643 − 1.19i)14-s + (0.523 + 0.483i)15-s + (0.392 + 0.919i)16-s + (−0.288 + 0.288i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.41729 + 0.341235i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41729 + 0.341235i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.70 - 0.813i)T \) |
| 5 | \( 1 + (0.435 + 11.1i)T \) |
good | 3 | \( 1 + (2.61 - 2.61i)T - 27iT^{2} \) |
| 7 | \( 1 + (17.7 + 17.7i)T + 343iT^{2} \) |
| 11 | \( 1 - 7.37iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (2.68 + 2.68i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 + (20.2 - 20.2i)T - 4.91e3iT^{2} \) |
| 19 | \( 1 - 135.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (71.0 - 71.0i)T - 1.21e4iT^{2} \) |
| 29 | \( 1 - 34.2iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 187. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-250. + 250. i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + 211.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (-46.7 + 46.7i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + (-189. - 189. i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 + (74.5 + 74.5i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 - 101.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 232.T + 2.26e5T^{2} \) |
| 67 | \( 1 + (34.7 + 34.7i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 - 614. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (37.4 + 37.4i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 + 1.00e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + (423. - 423. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + 1.04e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (536. - 536. i)T - 9.12e5iT^{2} \) |
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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
−17.28586556033312289397872614767, −16.37706990328811209678740952019, −15.77035175837558920781744992572, −13.80580975953522128164142216406, −12.91425202589678725531912227381, −11.47840074552586007040392685441, −9.886522970433804201978677728175, −7.56371739205221966535426462811, −5.61241536916767869930048852112, −4.10991926619355091410625798215,
3.06566795812459389820351851678, 5.87755051499836557867356258851, 6.90094636540833967842366730723, 9.826041882519395471945332774837, 11.51156679435831060116724522273, 12.33213665290293024797384334572, 13.73686238207739368742804246332, 15.09069164039564170599902342854, 16.11388004653613010191264512971, 18.15053734349966410797154992368