| L(s) = 1 | + (0.156 + 0.987i)2-s + (−1.94 + 0.992i)3-s + (−0.951 + 0.309i)4-s + (−2.03 − 0.919i)5-s + (−1.28 − 1.76i)6-s + (−1.55 + 3.05i)7-s + (−0.453 − 0.891i)8-s + (1.04 − 1.43i)9-s + (0.589 − 2.15i)10-s + (3.24 + 0.694i)11-s + (1.54 − 1.54i)12-s + (−2.44 + 0.387i)13-s + (−3.26 − 1.06i)14-s + (4.88 − 0.232i)15-s + (0.809 − 0.587i)16-s + (1.24 + 0.196i)17-s + ⋯ |
| L(s) = 1 | + (0.110 + 0.698i)2-s + (−1.12 + 0.572i)3-s + (−0.475 + 0.154i)4-s + (−0.911 − 0.411i)5-s + (−0.524 − 0.721i)6-s + (−0.588 + 1.15i)7-s + (−0.160 − 0.315i)8-s + (0.347 − 0.478i)9-s + (0.186 − 0.682i)10-s + (0.977 + 0.209i)11-s + (0.446 − 0.446i)12-s + (−0.679 + 0.107i)13-s + (−0.872 − 0.283i)14-s + (1.26 − 0.0600i)15-s + (0.202 − 0.146i)16-s + (0.301 + 0.0477i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.157i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0363780 + 0.458797i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0363780 + 0.458797i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.156 - 0.987i)T \) |
| 5 | \( 1 + (2.03 + 0.919i)T \) |
| 11 | \( 1 + (-3.24 - 0.694i)T \) |
| good | 3 | \( 1 + (1.94 - 0.992i)T + (1.76 - 2.42i)T^{2} \) |
| 7 | \( 1 + (1.55 - 3.05i)T + (-4.11 - 5.66i)T^{2} \) |
| 13 | \( 1 + (2.44 - 0.387i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-1.24 - 0.196i)T + (16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (1.62 - 5.00i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-5.42 - 5.42i)T + 23iT^{2} \) |
| 29 | \( 1 + (0.472 + 1.45i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (6.34 + 4.61i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (5.11 + 2.60i)T + (21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (-3.98 - 1.29i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (0.832 - 0.832i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.53 - 4.97i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (1.42 + 9.00i)T + (-50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (2.65 - 0.862i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-6.34 - 8.73i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (4.03 - 4.03i)T - 67iT^{2} \) |
| 71 | \( 1 + (7.25 - 5.27i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-8.12 - 4.14i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-11.2 - 8.19i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-1.89 + 11.9i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + 4.89iT - 89T^{2} \) |
| 97 | \( 1 + (-2.79 + 0.442i)T + (92.2 - 29.9i)T^{2} \) |
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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
−14.60730809771112970112569366709, −12.79006282882040433952219027273, −12.08187126418593601876331742109, −11.31644971088557344884958445900, −9.748633764783981262794526059003, −8.848253975965449707714563864591, −7.40699497695329574693962544387, −6.03793934143337535652212855072, −5.16527311753084041639198173897, −3.85326029995503657411285872048,
0.58106514391998670014830932882, 3.41439945213466856684578116349, 4.78851427395055422424243769795, 6.61560082616002044480401579399, 7.21436279128171489776402648069, 9.022392801880112589836864932716, 10.58974892751945683968000708017, 11.07006971260921074020277051231, 12.14302290835339053082316850847, 12.74176106484620900695196083438
