| 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
−12.74176106484620900695196083438, −12.14302290835339053082316850847, −11.07006971260921074020277051231, −10.58974892751945683968000708017, −9.022392801880112589836864932716, −7.21436279128171489776402648069, −6.61560082616002044480401579399, −4.78851427395055422424243769795, −3.41439945213466856684578116349, −0.58106514391998670014830932882,
3.85326029995503657411285872048, 5.16527311753084041639198173897, 6.03793934143337535652212855072, 7.40699497695329574693962544387, 8.848253975965449707714563864591, 9.748633764783981262794526059003, 11.31644971088557344884958445900, 12.08187126418593601876331742109, 12.79006282882040433952219027273, 14.60730809771112970112569366709
