L(s) = 1 | + (0.5 + 0.866i)3-s + (3.72 + 2.14i)5-s + (−0.499 + 0.866i)9-s + (4.38 − 2.53i)11-s − 3.37i·13-s + 4.29i·15-s + (−2.39 + 1.38i)17-s + (2.35 − 4.07i)19-s + (4.18 + 2.41i)23-s + (6.73 + 11.6i)25-s − 0.999·27-s + 2.46·29-s + (−2.84 − 4.93i)31-s + (4.38 + 2.53i)33-s + (1.16 − 2.02i)37-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (1.66 + 0.960i)5-s + (−0.166 + 0.288i)9-s + (1.32 − 0.763i)11-s − 0.937i·13-s + 1.10i·15-s + (−0.581 + 0.335i)17-s + (0.539 − 0.934i)19-s + (0.872 + 0.503i)23-s + (1.34 + 2.33i)25-s − 0.192·27-s + 0.456·29-s + (−0.511 − 0.886i)31-s + (0.763 + 0.440i)33-s + (0.191 − 0.332i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.769 - 0.638i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.048334905\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.048334905\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-3.72 - 2.14i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.38 + 2.53i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 3.37iT - 13T^{2} \) |
| 17 | \( 1 + (2.39 - 1.38i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.35 + 4.07i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.18 - 2.41i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2.46T + 29T^{2} \) |
| 31 | \( 1 + (2.84 + 4.93i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.16 + 2.02i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 5.14iT - 41T^{2} \) |
| 43 | \( 1 + 13.0iT - 43T^{2} \) |
| 47 | \( 1 + (-2.67 + 4.63i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.11 - 3.65i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.80 - 8.31i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.35 + 1.93i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.12 - 2.38i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12.3iT - 71T^{2} \) |
| 73 | \( 1 + (9.96 - 5.75i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (12.0 + 6.95i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 7.32T + 83T^{2} \) |
| 89 | \( 1 + (12.1 + 7.02i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 14.5iT - 97T^{2} \) |
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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
−9.044547355955440374631626374862, −8.730938336361570650320154738942, −7.28897881099899927274740763290, −6.74161768189157832163294801320, −5.78858471877983614951246087296, −5.45057426589535035807279457890, −4.10542922030655550400056567397, −3.12070394462379750066552118507, −2.49597675735239482941890933082, −1.25784962803439150522974216508,
1.31972399209735990385348819206, 1.70705599785366863835706219948, 2.81516048288219000684706095013, 4.26110266530773649750885072942, 4.90705523421814756244396877682, 5.91320261542287442276444009362, 6.56208324860114691750373065127, 7.13470495544919996543422632069, 8.410038719646430007942386029236, 9.028755179397459469314631923666