| L(s) = 1 | + (−0.525 + 2.47i)2-s + (0.803 + 0.892i)3-s + (−4.00 − 1.78i)4-s + 0.793i·5-s + (−2.62 + 1.51i)6-s + (−0.479 + 1.07i)7-s + (3.54 − 4.87i)8-s + (0.162 − 1.54i)9-s + (−1.96 − 0.416i)10-s + (−5.94 + 0.624i)11-s + (−1.62 − 5.00i)12-s + (−3.60 + 0.0625i)13-s + (−2.40 − 1.74i)14-s + (−0.708 + 0.637i)15-s + (4.31 + 4.79i)16-s + (−0.207 + 1.97i)17-s + ⋯ |
| L(s) = 1 | + (−0.371 + 1.74i)2-s + (0.464 + 0.515i)3-s + (−2.00 − 0.891i)4-s + 0.354i·5-s + (−1.07 + 0.619i)6-s + (−0.181 + 0.406i)7-s + (1.25 − 1.72i)8-s + (0.0542 − 0.516i)9-s + (−0.620 − 0.131i)10-s + (−1.79 + 0.188i)11-s + (−0.469 − 1.44i)12-s + (−0.999 + 0.0173i)13-s + (−0.643 − 0.467i)14-s + (−0.182 + 0.164i)15-s + (1.07 + 1.19i)16-s + (−0.0503 + 0.479i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.382957 - 0.463007i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.382957 - 0.463007i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (3.60 - 0.0625i)T \) |
| 31 | \( 1 + (4.74 - 2.91i)T \) |
| good | 2 | \( 1 + (0.525 - 2.47i)T + (-1.82 - 0.813i)T^{2} \) |
| 3 | \( 1 + (-0.803 - 0.892i)T + (-0.313 + 2.98i)T^{2} \) |
| 5 | \( 1 - 0.793iT - 5T^{2} \) |
| 7 | \( 1 + (0.479 - 1.07i)T + (-4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (5.94 - 0.624i)T + (10.7 - 2.28i)T^{2} \) |
| 17 | \( 1 + (0.207 - 1.97i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-2.08 - 1.87i)T + (1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (7.98 - 3.55i)T + (15.3 - 17.0i)T^{2} \) |
| 29 | \( 1 + (-2.01 - 0.428i)T + (26.4 + 11.7i)T^{2} \) |
| 37 | \( 1 + (-8.59 - 4.96i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.95 - 9.21i)T + (-37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-4.00 + 4.44i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (-2.00 - 0.652i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (3.27 + 2.37i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.09 - 9.86i)T + (-53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-0.117 - 0.202i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.39 + 3.68i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.39 + 0.147i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (6.87 + 9.46i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (1.58 + 1.15i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (5.89 - 1.91i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-11.6 + 1.22i)T + (87.0 - 18.5i)T^{2} \) |
| 97 | \( 1 + (2.50 - 5.62i)T + (-64.9 - 72.0i)T^{2} \) |
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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
−12.08459016311954069575237038537, −10.37548590831678790607497140691, −9.852955022794194573406714298163, −9.009214035641139060957985541474, −7.997309855232941475750118044309, −7.47773492851310735561586732881, −6.29253725882779887385776080363, −5.44867415040478886767614942851, −4.44401492892550064462961618847, −2.87580762932701286301711028106,
0.39952500223723525814041496516, 2.22849076101009914943545567653, 2.79117892268689793059153970443, 4.33455060560237649673972499149, 5.30338755261365517174988740703, 7.39725350057222269065840968942, 8.022440993490407811582984322051, 8.978630269619670041156952631487, 10.02061467334529955224292920895, 10.52228795507478163499204394150
