| L(s) = 1 | + (−2.39 − 1.06i)2-s + (−0.206 + 1.96i)3-s + (3.26 + 3.62i)4-s + 1.57·5-s + (2.58 − 4.48i)6-s + (−2.63 − 2.92i)7-s + (−2.33 − 7.17i)8-s + (−0.876 − 0.186i)9-s + (−3.76 − 1.67i)10-s + (−1.71 + 0.364i)11-s + (−7.78 + 5.65i)12-s + (−2.82 − 2.24i)13-s + (3.18 + 9.80i)14-s + (−0.324 + 3.08i)15-s + (−1.04 + 9.97i)16-s + (0.480 + 0.102i)17-s + ⋯ |
| L(s) = 1 | + (−1.69 − 0.754i)2-s + (−0.119 + 1.13i)3-s + (1.63 + 1.81i)4-s + 0.703·5-s + (1.05 − 1.83i)6-s + (−0.994 − 1.10i)7-s + (−0.824 − 2.53i)8-s + (−0.292 − 0.0621i)9-s + (−1.19 − 0.530i)10-s + (−0.517 + 0.110i)11-s + (−2.24 + 1.63i)12-s + (−0.782 − 0.622i)13-s + (0.851 + 2.62i)14-s + (−0.0838 + 0.797i)15-s + (−0.262 + 2.49i)16-s + (0.116 + 0.0247i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.246186 - 0.302059i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.246186 - 0.302059i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (2.82 + 2.24i)T \) |
| 31 | \( 1 + (5.54 - 0.447i)T \) |
| good | 2 | \( 1 + (2.39 + 1.06i)T + (1.33 + 1.48i)T^{2} \) |
| 3 | \( 1 + (0.206 - 1.96i)T + (-2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 - 1.57T + 5T^{2} \) |
| 7 | \( 1 + (2.63 + 2.92i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (1.71 - 0.364i)T + (10.0 - 4.47i)T^{2} \) |
| 17 | \( 1 + (-0.480 - 0.102i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (0.652 + 6.20i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-5.24 + 5.82i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-3.02 - 1.34i)T + (19.4 + 21.5i)T^{2} \) |
| 37 | \( 1 + (1.36 + 2.35i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.76 - 3.45i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (0.708 + 6.73i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (3.01 + 2.18i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (3.03 + 9.35i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-6.88 + 3.06i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-5.89 + 10.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.59 - 4.49i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (7.15 + 1.52i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (-0.530 + 1.63i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (1.65 + 5.10i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (7.55 - 5.48i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-1.35 + 0.287i)T + (81.3 - 36.1i)T^{2} \) |
| 97 | \( 1 + (-3.19 - 3.55i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−10.58204899883195192683835561566, −10.04658613935898206340166740213, −9.573636345795410030969219953527, −8.749696691521563110646932029612, −7.42071523205074185145737793508, −6.73404337112649581136398917851, −4.97933777039449864721639631301, −3.54609593987477604141498032676, −2.49120892656193393004151331548, −0.43668513532401899949064364666,
1.53147514184995265870093753532, 2.54495468080057300478941508023, 5.67571474798852600029625979107, 6.07034520212172723794536143847, 7.06952916965889588990929078202, 7.71243867377353163534324595026, 8.788091250181881322068230881414, 9.598918651687313792286721804898, 10.05091977242950995794365005005, 11.39617829908300672888254412189
