| L(s) = 1 | + (1.37 + 0.348i)2-s + (−1.57 + 0.719i)3-s + (1.75 + 0.955i)4-s + (−2.68 + 2.32i)5-s + (−2.40 + 0.436i)6-s + (0.314 + 0.202i)7-s + (2.07 + 1.92i)8-s + (−0.00255 + 0.00294i)9-s + (−4.48 + 2.24i)10-s + (3.35 − 0.482i)11-s + (−3.45 − 0.240i)12-s + (−0.0151 − 0.0234i)13-s + (0.360 + 0.386i)14-s + (2.55 − 5.58i)15-s + (2.17 + 3.35i)16-s + (−0.162 + 0.0478i)17-s + ⋯ |
| L(s) = 1 | + (0.969 + 0.246i)2-s + (−0.909 + 0.415i)3-s + (0.878 + 0.477i)4-s + (−1.19 + 1.03i)5-s + (−0.983 + 0.178i)6-s + (0.118 + 0.0764i)7-s + (0.733 + 0.679i)8-s + (−0.000852 + 0.000983i)9-s + (−1.41 + 0.711i)10-s + (1.01 − 0.145i)11-s + (−0.996 − 0.0694i)12-s + (−0.00418 − 0.00651i)13-s + (0.0964 + 0.103i)14-s + (0.658 − 1.44i)15-s + (0.543 + 0.839i)16-s + (−0.0395 + 0.0115i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.862135 + 1.01744i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.862135 + 1.01744i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.37 - 0.348i)T \) |
| 23 | \( 1 + (-4.32 + 2.07i)T \) |
| good | 3 | \( 1 + (1.57 - 0.719i)T + (1.96 - 2.26i)T^{2} \) |
| 5 | \( 1 + (2.68 - 2.32i)T + (0.711 - 4.94i)T^{2} \) |
| 7 | \( 1 + (-0.314 - 0.202i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-3.35 + 0.482i)T + (10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (0.0151 + 0.0234i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (0.162 - 0.0478i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-1.16 + 3.97i)T + (-15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-2.21 - 7.54i)T + (-24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (1.41 - 3.10i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (8.60 + 7.45i)T + (5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-4.85 - 5.60i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-9.01 + 4.11i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 5.96T + 47T^{2} \) |
| 53 | \( 1 + (0.748 - 1.16i)T + (-22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-1.06 - 1.66i)T + (-24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-7.17 - 3.27i)T + (39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (7.10 + 1.02i)T + (64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (0.00700 - 0.0486i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-11.6 - 3.42i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (5.97 - 3.84i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (4.52 + 3.91i)T + (11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (6.81 + 14.9i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (3.32 + 3.84i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−12.65144007042309178570856613759, −11.75182709427459239243211981837, −11.15119021924079324823063559754, −10.61906951963378369490667361727, −8.667939168775471669113317258678, −7.24492016678756816449793670019, −6.63800648528115229721361348932, −5.29767072467474930739901485530, −4.20507760912412174291250063138, −3.05995412077044611619191686755,
1.12435611509793597769433496357, 3.67600504055977296915738320112, 4.67601678435815521996513184034, 5.78415146948827476106395883761, 6.89157608184970897447566333201, 8.011815223109686998750129761907, 9.435611885853450517797391588545, 11.01077541810306091118525332991, 11.75078199749619414525379108524, 12.18368614057671127679479162639
