L(s) = 1 | + (0.453 − 0.891i)2-s + (−0.443 − 2.79i)3-s + (−0.587 − 0.809i)4-s + (−1.71 + 1.44i)5-s + (−2.69 − 0.875i)6-s + (3.26 + 0.516i)7-s + (−0.987 + 0.156i)8-s + (−4.78 + 1.55i)9-s + (0.506 + 2.17i)10-s + (−0.266 − 3.30i)11-s + (−2.00 + 2.00i)12-s + (2.49 + 1.27i)13-s + (1.94 − 2.67i)14-s + (4.78 + 4.14i)15-s + (−0.309 + 0.951i)16-s + (2.57 − 1.30i)17-s + ⋯ |
L(s) = 1 | + (0.321 − 0.630i)2-s + (−0.255 − 1.61i)3-s + (−0.293 − 0.404i)4-s + (−0.764 + 0.644i)5-s + (−1.10 − 0.357i)6-s + (1.23 + 0.195i)7-s + (−0.349 + 0.0553i)8-s + (−1.59 + 0.518i)9-s + (0.160 + 0.688i)10-s + (−0.0803 − 0.996i)11-s + (−0.578 + 0.578i)12-s + (0.692 + 0.353i)13-s + (0.518 − 0.713i)14-s + (1.23 + 1.07i)15-s + (−0.0772 + 0.237i)16-s + (0.623 − 0.317i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.511 + 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.511 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.524206 - 0.921719i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.524206 - 0.921719i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.453 + 0.891i)T \) |
| 5 | \( 1 + (1.71 - 1.44i)T \) |
| 11 | \( 1 + (0.266 + 3.30i)T \) |
good | 3 | \( 1 + (0.443 + 2.79i)T + (-2.85 + 0.927i)T^{2} \) |
| 7 | \( 1 + (-3.26 - 0.516i)T + (6.65 + 2.16i)T^{2} \) |
| 13 | \( 1 + (-2.49 - 1.27i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-2.57 + 1.30i)T + (9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (0.213 + 0.155i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-3.16 - 3.16i)T + 23iT^{2} \) |
| 29 | \( 1 + (6.91 - 5.02i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.80 - 8.63i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.251 - 1.58i)T + (-35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (2.34 - 3.22i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-0.539 + 0.539i)T - 43iT^{2} \) |
| 47 | \( 1 + (6.91 - 1.09i)T + (44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-3.38 + 6.64i)T + (-31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-1.61 - 2.22i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (2.35 + 0.763i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (1.62 - 1.62i)T - 67iT^{2} \) |
| 71 | \( 1 + (-3.37 + 10.3i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.297 - 1.88i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (4.10 + 12.6i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (0.378 + 0.743i)T + (-48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 - 17.3iT - 89T^{2} \) |
| 97 | \( 1 + (-4.57 - 2.33i)T + (57.0 + 78.4i)T^{2} \) |
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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
−13.26913057562506234560406131628, −12.08126674120990335626077442021, −11.44171087136919701511398331368, −10.86355172339269833962541292464, −8.640573086932676782608207995690, −7.77300071267821978158811759745, −6.63884852360635083871112304521, −5.28358160145063299063088656916, −3.23906436422842620729365106619, −1.44359799081257878435217138776,
3.92066349295055219683339004772, 4.62412727972157031088563004401, 5.57752346442920088817424986159, 7.64421311355892131989206797486, 8.548878049681955704748089922636, 9.716220868041262171436591406002, 10.94392507029352242649465611662, 11.75061611234279939851984906046, 13.03772969372729645762370247492, 14.56726201623800865979853669453