| L(s) = 1 | + (0.984 + 0.173i)2-s + (1.37 − 1.05i)3-s + (0.939 + 0.342i)4-s + (1.94 + 0.706i)5-s + (1.53 − 0.799i)6-s + (−2.54 − 0.729i)7-s + (0.866 + 0.5i)8-s + (0.777 − 2.89i)9-s + (1.78 + 1.03i)10-s + (0.520 + 1.43i)11-s + (1.65 − 0.520i)12-s + (−0.512 + 1.40i)13-s + (−2.37 − 1.15i)14-s + (3.41 − 1.07i)15-s + (0.766 + 0.642i)16-s + (1.09 − 1.88i)17-s + ⋯ |
| L(s) = 1 | + (0.696 + 0.122i)2-s + (0.793 − 0.608i)3-s + (0.469 + 0.171i)4-s + (0.868 + 0.316i)5-s + (0.627 − 0.326i)6-s + (−0.961 − 0.275i)7-s + (0.306 + 0.176i)8-s + (0.259 − 0.965i)9-s + (0.565 + 0.326i)10-s + (0.157 + 0.431i)11-s + (0.476 − 0.150i)12-s + (−0.142 + 0.390i)13-s + (−0.635 − 0.309i)14-s + (0.881 − 0.277i)15-s + (0.191 + 0.160i)16-s + (0.264 − 0.458i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.61801 - 0.324143i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.61801 - 0.324143i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.984 - 0.173i)T \) |
| 3 | \( 1 + (-1.37 + 1.05i)T \) |
| 7 | \( 1 + (2.54 + 0.729i)T \) |
| good | 5 | \( 1 + (-1.94 - 0.706i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (-0.520 - 1.43i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (0.512 - 1.40i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.09 + 1.88i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.34 - 0.778i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.19 - 0.916i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.292 - 0.802i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (2.46 - 6.76i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 - 9.89T + 37T^{2} \) |
| 41 | \( 1 + (9.51 + 3.46i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.210 + 1.19i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (3.39 - 1.23i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-4.42 + 2.55i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.56 + 2.99i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (1.69 + 4.65i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.06 - 6.03i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (4.04 - 2.33i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 9.34iT - 73T^{2} \) |
| 79 | \( 1 + (1.42 - 8.09i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (9.24 - 3.36i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (9.30 + 16.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.71 + 1.18i)T + (91.1 + 33.1i)T^{2} \) |
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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
−11.63339274744318785052920440986, −10.11375377771705173632612298352, −9.682304936676175988270163789448, −8.485899420150892861989154838964, −7.22903244897434315097672261805, −6.63665656660240314197941677479, −5.73817389489231362629252845711, −4.11744110835109387594626344725, −3.01256139157467541406918233811, −1.90938075320979788133426905227,
2.12025917565163736025525523801, 3.22070792427650337858829000950, 4.27631008108523049970781737017, 5.58630582822978535671126310229, 6.26178522145222220462101662142, 7.74007811175071727614240243367, 8.814818313782305944894667727518, 9.767024047281578691560920819753, 10.19294956980759387357063817803, 11.41527773806127059950033892790
