L(s) = 1 | + (0.794 + 1.37i)2-s + (1.04 − 1.37i)3-s + (−0.262 + 0.454i)4-s + (−1.14 + 0.662i)5-s + (2.73 + 0.348i)6-s + (2.48 − 0.904i)7-s + 2.34·8-s + (−0.798 − 2.89i)9-s + (−1.82 − 1.05i)10-s + (−3.17 − 0.964i)11-s + (0.351 + 0.839i)12-s + 3.14i·13-s + (3.22 + 2.70i)14-s + (−0.290 + 2.27i)15-s + (2.38 + 4.13i)16-s + (−1.28 + 2.22i)17-s + ⋯ |
L(s) = 1 | + (0.561 + 0.973i)2-s + (0.605 − 0.795i)3-s + (−0.131 + 0.227i)4-s + (−0.513 + 0.296i)5-s + (1.11 + 0.142i)6-s + (0.939 − 0.341i)7-s + 0.828·8-s + (−0.266 − 0.963i)9-s + (−0.576 − 0.333i)10-s + (−0.956 − 0.290i)11-s + (0.101 + 0.242i)12-s + 0.870i·13-s + (0.860 + 0.722i)14-s + (−0.0751 + 0.588i)15-s + (0.596 + 1.03i)16-s + (−0.312 + 0.540i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.91589 + 0.375193i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.91589 + 0.375193i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.04 + 1.37i)T \) |
| 7 | \( 1 + (-2.48 + 0.904i)T \) |
| 11 | \( 1 + (3.17 + 0.964i)T \) |
good | 2 | \( 1 + (-0.794 - 1.37i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (1.14 - 0.662i)T + (2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 3.14iT - 13T^{2} \) |
| 17 | \( 1 + (1.28 - 2.22i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.54 - 2.04i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.23 + 3.59i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4.92T + 29T^{2} \) |
| 31 | \( 1 + (0.543 - 0.940i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.551 + 0.955i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6.61T + 41T^{2} \) |
| 43 | \( 1 + 5.86iT - 43T^{2} \) |
| 47 | \( 1 + (1.18 - 0.685i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.25 - 1.30i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (9.38 + 5.41i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (8.35 - 4.82i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.67 + 8.09i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 11.4iT - 71T^{2} \) |
| 73 | \( 1 + (-11.3 - 6.55i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.91 + 3.41i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 + (-11.1 + 6.41i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6.94T + 97T^{2} \) |
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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
−12.61078300946073108167775488029, −11.29971882357897397462462629220, −10.58517568979879242692485289276, −8.810078234601007671312426754126, −7.934834553872229144750013408187, −7.25682843106846414875848128144, −6.37225960461013236083364932196, −5.05762472723726379070253307882, −3.81099443756697979721307900205, −1.90554761591351372296917865559,
2.22503832917085154124080513737, 3.31580072287920880801290387363, 4.62413946543658500127249805588, 5.15810842755909097106194055923, 7.58169344560583686362271564258, 8.188499920442566538780616816183, 9.366211617224770254315859035944, 10.64209452206507650980784909847, 11.05583434008915981786400552631, 12.06377372550138006779873226014