| 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
−14.56726201623800865979853669453, −13.03772969372729645762370247492, −11.75061611234279939851984906046, −10.94392507029352242649465611662, −9.716220868041262171436591406002, −8.548878049681955704748089922636, −7.64421311355892131989206797486, −5.57752346442920088817424986159, −4.62412727972157031088563004401, −3.92066349295055219683339004772,
1.44359799081257878435217138776, 3.23906436422842620729365106619, 5.28358160145063299063088656916, 6.63884852360635083871112304521, 7.77300071267821978158811759745, 8.640573086932676782608207995690, 10.86355172339269833962541292464, 11.44171087136919701511398331368, 12.08126674120990335626077442021, 13.26913057562506234560406131628
