| L(s) = 1 | + (−1.53 − 0.797i)3-s + (0.258 − 0.710i)5-s + (−0.777 + 1.34i)7-s + (1.72 + 2.45i)9-s + (0.832 − 0.480i)11-s + (0.416 − 0.496i)13-s + (−0.963 + 0.886i)15-s + (−6.73 + 1.18i)17-s + (4.14 + 1.35i)19-s + (2.27 − 1.45i)21-s + (0.400 + 1.10i)23-s + (3.39 + 2.84i)25-s + (−0.705 − 5.14i)27-s + (−1.39 + 7.92i)29-s + (2.63 + 1.52i)31-s + ⋯ |
| L(s) = 1 | + (−0.887 − 0.460i)3-s + (0.115 − 0.317i)5-s + (−0.294 + 0.509i)7-s + (0.576 + 0.817i)9-s + (0.250 − 0.144i)11-s + (0.115 − 0.137i)13-s + (−0.248 + 0.228i)15-s + (−1.63 + 0.287i)17-s + (0.950 + 0.310i)19-s + (0.495 − 0.316i)21-s + (0.0835 + 0.229i)23-s + (0.678 + 0.569i)25-s + (−0.135 − 0.990i)27-s + (−0.259 + 1.47i)29-s + (0.474 + 0.273i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01999 + 0.234064i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01999 + 0.234064i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.53 + 0.797i)T \) |
| 19 | \( 1 + (-4.14 - 1.35i)T \) |
| good | 5 | \( 1 + (-0.258 + 0.710i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.777 - 1.34i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.832 + 0.480i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.416 + 0.496i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (6.73 - 1.18i)T + (15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-0.400 - 1.10i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (1.39 - 7.92i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-2.63 - 1.52i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4.12iT - 37T^{2} \) |
| 41 | \( 1 + (-4.09 + 3.43i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-7.34 - 2.67i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (3.11 + 0.548i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-13.6 + 4.96i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-2.02 - 11.4i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-10.1 + 3.70i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-9.19 - 1.62i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-0.0322 - 0.0117i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (3.04 - 2.55i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-0.893 - 1.06i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (10.4 + 6.05i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (4.68 + 3.92i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-9.54 + 1.68i)T + (91.1 - 33.1i)T^{2} \) |
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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
−10.31127612002078018127458005178, −9.198720852502041325966998527714, −8.626261636363474759158194157600, −7.37412493573990722150332700382, −6.72230859666668098108980472078, −5.75209459205521535103726291948, −5.13294512512608337373377230001, −3.96477117438654133148017448627, −2.46521277478285173047360518553, −1.12435648534947389083152868973,
0.67753119285854557840859223650, 2.53817592873860931818991932076, 3.95703567490805275733493383179, 4.60950529634515622636487606600, 5.73505277880464536232113394406, 6.66327120335579518389338824970, 7.10692994021459361865312386129, 8.467329740232549927497238410153, 9.504958349319519468569255870365, 10.00295652373004070282673151811
