| L(s) = 1 | + (1.40 + 0.181i)2-s + (0.995 + 0.0980i)3-s + (1.93 + 0.509i)4-s + (1.31 + 2.46i)5-s + (1.37 + 0.318i)6-s + (−0.901 − 4.53i)7-s + (2.62 + 1.06i)8-s + (0.980 + 0.195i)9-s + (1.39 + 3.69i)10-s + (−3.61 − 2.96i)11-s + (1.87 + 0.696i)12-s + (−0.706 − 0.377i)13-s + (−0.441 − 6.52i)14-s + (1.06 + 2.58i)15-s + (3.48 + 1.96i)16-s + (−2.89 + 6.98i)17-s + ⋯ |
| L(s) = 1 | + (0.991 + 0.128i)2-s + (0.574 + 0.0565i)3-s + (0.967 + 0.254i)4-s + (0.588 + 1.10i)5-s + (0.562 + 0.129i)6-s + (−0.340 − 1.71i)7-s + (0.926 + 0.376i)8-s + (0.326 + 0.0650i)9-s + (0.442 + 1.16i)10-s + (−1.09 − 0.894i)11-s + (0.541 + 0.201i)12-s + (−0.195 − 0.104i)13-s + (−0.118 − 1.74i)14-s + (0.275 + 0.666i)15-s + (0.870 + 0.492i)16-s + (−0.701 + 1.69i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.89030 + 0.475442i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.89030 + 0.475442i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 - 0.181i)T \) |
| 3 | \( 1 + (-0.995 - 0.0980i)T \) |
| good | 5 | \( 1 + (-1.31 - 2.46i)T + (-2.77 + 4.15i)T^{2} \) |
| 7 | \( 1 + (0.901 + 4.53i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (3.61 + 2.96i)T + (2.14 + 10.7i)T^{2} \) |
| 13 | \( 1 + (0.706 + 0.377i)T + (7.22 + 10.8i)T^{2} \) |
| 17 | \( 1 + (2.89 - 6.98i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.380 - 0.115i)T + (15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (5.16 - 3.45i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (-0.0117 - 0.0143i)T + (-5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (0.0486 - 0.0486i)T - 31iT^{2} \) |
| 37 | \( 1 + (-1.34 + 4.42i)T + (-30.7 - 20.5i)T^{2} \) |
| 41 | \( 1 + (3.57 + 5.35i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (3.24 - 0.319i)T + (42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (-9.89 - 4.09i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-9.12 + 11.1i)T + (-10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (4.37 - 2.33i)T + (32.7 - 49.0i)T^{2} \) |
| 61 | \( 1 + (-0.556 + 5.65i)T + (-59.8 - 11.9i)T^{2} \) |
| 67 | \( 1 + (-0.0180 + 0.182i)T + (-65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (-0.279 + 0.0555i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (2.11 - 10.6i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-0.948 + 0.392i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-0.874 - 2.88i)T + (-69.0 + 46.1i)T^{2} \) |
| 89 | \( 1 + (-10.5 - 7.06i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (4.10 - 4.10i)T - 97iT^{2} \) |
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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
−11.10199914919465614667477448371, −10.51932445316904778570737285028, −10.10376258948480044569898360464, −8.252434833965927864531287030471, −7.42227757879690533171750742946, −6.60353093376745185189926773711, −5.70279660015578062775606422556, −4.08255871481941007973491513133, −3.40238316873383386129067365893, −2.16844439061453754316246230696,
2.13375917789289864260544280679, 2.71999758086060005787531508552, 4.64560411499826662773509881823, 5.20860578194613960283027564545, 6.21119993976306226020972944359, 7.47671879670048909443015930619, 8.670305997649499284461017544250, 9.425608756294416450834553002539, 10.26441910606631963007930226302, 11.89230054029699454604585890486
