| L(s) = 1 | + (−0.173 − 0.984i)2-s + (−0.716 + 1.57i)3-s + (−0.939 + 0.342i)4-s + (−1.14 + 3.13i)5-s + (1.67 + 0.431i)6-s + (−1.07 + 1.85i)7-s + (0.5 + 0.866i)8-s + (−1.97 − 2.25i)9-s + (3.28 + 0.579i)10-s + (5.41 − 3.12i)11-s + (0.133 − 1.72i)12-s + (−2.56 + 3.05i)13-s + (2.01 + 0.734i)14-s + (−4.12 − 4.04i)15-s + (0.766 − 0.642i)16-s + (0.403 − 0.0711i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (−0.413 + 0.910i)3-s + (−0.469 + 0.171i)4-s + (−0.510 + 1.40i)5-s + (0.684 + 0.176i)6-s + (−0.405 + 0.702i)7-s + (0.176 + 0.306i)8-s + (−0.657 − 0.753i)9-s + (1.03 + 0.183i)10-s + (1.63 − 0.943i)11-s + (0.0386 − 0.498i)12-s + (−0.710 + 0.846i)13-s + (0.539 + 0.196i)14-s + (−1.06 − 1.04i)15-s + (0.191 − 0.160i)16-s + (0.0978 − 0.0172i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.583721 + 0.428426i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.583721 + 0.428426i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 3 | \( 1 + (0.716 - 1.57i)T \) |
| 19 | \( 1 + (-4.34 - 0.329i)T \) |
| good | 5 | \( 1 + (1.14 - 3.13i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (1.07 - 1.85i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.41 + 3.12i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.56 - 3.05i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.403 + 0.0711i)T + (15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-0.280 - 0.770i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.805 - 4.56i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (2.02 + 1.16i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6.01iT - 37T^{2} \) |
| 41 | \( 1 + (0.926 - 0.777i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-5.87 - 2.13i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-7.59 - 1.33i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-0.220 + 0.0802i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (0.930 + 5.27i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-7.30 + 2.65i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-3.48 - 0.614i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (4.19 + 1.52i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (4.33 - 3.63i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (8.05 + 9.59i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (8.01 + 4.62i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.61 - 4.71i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (16.0 - 2.83i)T + (91.1 - 33.1i)T^{2} \) |
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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
−14.23878017846663438990138471630, −12.19366049780794600448780063662, −11.53017458114950901032136230284, −10.90150076850886435691162273932, −9.627849257385755143341914270314, −8.969974393012676876302076874259, −7.04947411258760509677355123068, −5.83882045014187971760890101473, −3.98490243336848218158546096221, −3.04200396488313173551979424880,
0.968620835113525867493081073357, 4.24973447452945807905841390481, 5.47077155111520786637344021101, 6.92181678356832412710955437307, 7.67691032539036819522954550662, 8.862366960867335392392588714411, 9.957662599932164229121731476317, 11.77276660825422237421408402692, 12.41500899938295597057950190078, 13.27059515121763623995088490959
