L(s) = 1 | + (2 + 1.73i)7-s + 5.19i·13-s + (−3 + 1.73i)19-s + (2.5 − 4.33i)25-s + (7.5 + 4.33i)31-s + (5.5 + 9.52i)37-s − 5·43-s + (1.00 + 6.92i)49-s + (7.5 − 4.33i)61-s + (−2.5 + 4.33i)67-s + (12 + 6.92i)73-s + (−8.5 − 14.7i)79-s + (−9 + 10.3i)91-s − 19.0i·97-s + (−13.5 + 7.79i)103-s + ⋯ |
L(s) = 1 | + (0.755 + 0.654i)7-s + 1.44i·13-s + (−0.688 + 0.397i)19-s + (0.5 − 0.866i)25-s + (1.34 + 0.777i)31-s + (0.904 + 1.56i)37-s − 0.762·43-s + (0.142 + 0.989i)49-s + (0.960 − 0.554i)61-s + (−0.305 + 0.529i)67-s + (1.40 + 0.810i)73-s + (−0.956 − 1.65i)79-s + (−0.943 + 1.08i)91-s − 1.93i·97-s + (−1.33 + 0.767i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.38308 + 0.741119i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.38308 + 0.741119i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
good | 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5.19iT - 13T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + (-7.5 - 4.33i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.5 - 9.52i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.5 + 4.33i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-12 - 6.92i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8.5 + 14.7i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 19.0iT - 97T^{2} \) |
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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
−10.46207508259370605505636992102, −9.610307082363703982920359378759, −8.599620532316259353537985511860, −8.179722233185757838154967605616, −6.85283182151070359654774184826, −6.18460482440979899993377330770, −4.92986125229453471872497558453, −4.25792008117648876543269793641, −2.72508431959757112345433782391, −1.58818743037637945533912840598,
0.869840518111838942871331905076, 2.47168693844004023704618418888, 3.75286593814605939011564422898, 4.78544808537332050285433725828, 5.66803436863119794497106666484, 6.79978912344790554746277743852, 7.75945588857778625781650398968, 8.304684175124121524246268673264, 9.401251789733498716145497169872, 10.38273259269943048516489025902