| L(s) = 1 | + (2.30 + 0.675i)2-s + (−0.654 + 0.755i)3-s + (3.15 + 2.02i)4-s + (−0.195 + 1.35i)5-s + (−2.01 + 1.29i)6-s + (0.415 + 0.909i)7-s + (2.74 + 3.17i)8-s + (−0.142 − 0.989i)9-s + (−1.36 + 2.99i)10-s + (0.857 − 0.251i)11-s + (−3.59 + 1.05i)12-s + (−0.561 + 1.22i)13-s + (0.341 + 2.37i)14-s + (−0.898 − 1.03i)15-s + (1.06 + 2.33i)16-s + (−0.841 + 0.540i)17-s + ⋯ |
| L(s) = 1 | + (1.62 + 0.477i)2-s + (−0.378 + 0.436i)3-s + (1.57 + 1.01i)4-s + (−0.0873 + 0.607i)5-s + (−0.823 + 0.529i)6-s + (0.157 + 0.343i)7-s + (0.971 + 1.12i)8-s + (−0.0474 − 0.329i)9-s + (−0.432 + 0.946i)10-s + (0.258 − 0.0759i)11-s + (−1.03 + 0.304i)12-s + (−0.155 + 0.341i)13-s + (0.0912 + 0.634i)14-s + (−0.232 − 0.267i)15-s + (0.266 + 0.582i)16-s + (−0.204 + 0.131i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0174 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0174 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.12604 + 2.16339i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.12604 + 2.16339i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.654 - 0.755i)T \) |
| 7 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (-2.01 + 4.35i)T \) |
| good | 2 | \( 1 + (-2.30 - 0.675i)T + (1.68 + 1.08i)T^{2} \) |
| 5 | \( 1 + (0.195 - 1.35i)T + (-4.79 - 1.40i)T^{2} \) |
| 11 | \( 1 + (-0.857 + 0.251i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (0.561 - 1.22i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (0.841 - 0.540i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (1.12 + 0.724i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-0.315 + 0.202i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-1.30 - 1.51i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.0448 - 0.312i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.446 + 3.10i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-1.23 + 1.42i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 0.861T + 47T^{2} \) |
| 53 | \( 1 + (2.90 + 6.36i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-3.58 + 7.85i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (7.04 + 8.12i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-2.50 - 0.734i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-7.44 - 2.18i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-3.63 - 2.33i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (4.98 - 10.9i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.115 - 0.805i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (7.93 - 9.16i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (0.986 - 6.86i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−11.32983136738539879858666046604, −10.78565886138091407004956487302, −9.505893147772339395569330311999, −8.309137913260884252415985747717, −6.92897874985449315782593379670, −6.50122931341358585661235132251, −5.40648820923881863358828682626, −4.60783394069725688189881375640, −3.63074626199866128082963027832, −2.52959525905039258640120276577,
1.36649185510651833418458994319, 2.81925897522695618387143399784, 4.11411860690686824473590141875, 4.91597665384276647002353649226, 5.77378529946471130022565538593, 6.72365740842592445201681163419, 7.74950603220230800729274529843, 9.012383010026039828005475004232, 10.33298765830516472586943597089, 11.14754119638733553057125113290
