L(s) = 1 | + (0.939 + 0.342i)3-s + (0.494 + 2.80i)5-s + (−4.05 − 2.33i)7-s + (0.766 + 0.642i)9-s + (−4.23 + 2.44i)11-s + (−0.974 − 2.67i)13-s + (−0.494 + 2.80i)15-s + (−3.09 + 2.59i)17-s + (3.35 − 2.78i)19-s + (−3.00 − 3.58i)21-s + (−5.98 − 1.05i)23-s + (−2.92 + 1.06i)25-s + (0.500 + 0.866i)27-s + (−3.64 + 4.34i)29-s + (−3.14 + 5.44i)31-s + ⋯ |
L(s) = 1 | + (0.542 + 0.197i)3-s + (0.221 + 1.25i)5-s + (−1.53 − 0.884i)7-s + (0.255 + 0.214i)9-s + (−1.27 + 0.737i)11-s + (−0.270 − 0.742i)13-s + (−0.127 + 0.724i)15-s + (−0.751 + 0.630i)17-s + (0.770 − 0.637i)19-s + (−0.656 − 0.782i)21-s + (−1.24 − 0.220i)23-s + (−0.585 + 0.213i)25-s + (0.0962 + 0.166i)27-s + (−0.676 + 0.806i)29-s + (−0.564 + 0.977i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00146182 + 0.424379i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00146182 + 0.424379i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.939 - 0.342i)T \) |
| 19 | \( 1 + (-3.35 + 2.78i)T \) |
good | 5 | \( 1 + (-0.494 - 2.80i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (4.05 + 2.33i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (4.23 - 2.44i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.974 + 2.67i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (3.09 - 2.59i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (5.98 + 1.05i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (3.64 - 4.34i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (3.14 - 5.44i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.42iT - 37T^{2} \) |
| 41 | \( 1 + (2.01 - 5.54i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1.67 - 0.296i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-5.46 + 6.50i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-7.07 - 1.24i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (3.61 - 3.03i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.715 + 4.05i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.79 - 2.34i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (0.152 + 0.864i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (9.38 + 3.41i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (12.2 + 4.45i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-4.28 - 2.47i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.03 - 8.34i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-11.7 - 14.0i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−10.42438022842377030409218441086, −9.941705150759973618079195693759, −9.024039263062103720663417330736, −7.69308461036029229236570645556, −7.18083206264336222391869883221, −6.46424290956955652372207257148, −5.32066330895264457279213315867, −3.92863080226801658711831030801, −3.12110798433180373541415220949, −2.36212348215619867786148787871,
0.16808237980618150799309773148, 2.08124356048498963719876065814, 3.03281635995617289414137855244, 4.21645803366936592204650431681, 5.50317748162536754326000963634, 5.97505161021106525990671499126, 7.21925252073010475728738608601, 8.193001119735962451909958938503, 8.933373354350749898026198525203, 9.521072695239859917449003754230