| L(s) = 1 | + (0.654 − 0.514i)2-s + (0.814 − 0.580i)3-s + (−0.307 + 1.26i)4-s + (2.89 + 0.557i)5-s + (0.234 − 0.799i)6-s + (−0.647 − 2.56i)7-s + (1.14 + 2.50i)8-s + (0.327 − 0.945i)9-s + (2.17 − 1.12i)10-s + (−0.0363 + 0.0462i)11-s + (0.485 + 1.21i)12-s + (0.922 − 1.43i)13-s + (−1.74 − 1.34i)14-s + (2.67 − 1.22i)15-s + (−0.282 − 0.145i)16-s + (−3.00 + 2.86i)17-s + ⋯ |
| L(s) = 1 | + (0.463 − 0.364i)2-s + (0.470 − 0.334i)3-s + (−0.153 + 0.634i)4-s + (1.29 + 0.249i)5-s + (0.0958 − 0.326i)6-s + (−0.244 − 0.969i)7-s + (0.404 + 0.885i)8-s + (0.109 − 0.315i)9-s + (0.689 − 0.355i)10-s + (−0.0109 + 0.0139i)11-s + (0.140 + 0.350i)12-s + (0.255 − 0.398i)13-s + (−0.466 − 0.359i)14-s + (0.691 − 0.315i)15-s + (−0.0706 − 0.0364i)16-s + (−0.727 + 0.693i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.37159 - 0.383944i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.37159 - 0.383944i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.814 + 0.580i)T \) |
| 7 | \( 1 + (0.647 + 2.56i)T \) |
| 23 | \( 1 + (-2.37 + 4.16i)T \) |
| good | 2 | \( 1 + (-0.654 + 0.514i)T + (0.471 - 1.94i)T^{2} \) |
| 5 | \( 1 + (-2.89 - 0.557i)T + (4.64 + 1.85i)T^{2} \) |
| 11 | \( 1 + (0.0363 - 0.0462i)T + (-2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-0.922 + 1.43i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (3.00 - 2.86i)T + (0.808 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-3.24 - 3.09i)T + (0.904 + 18.9i)T^{2} \) |
| 29 | \( 1 + (-3.59 - 1.05i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (0.361 - 3.78i)T + (-30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (6.60 + 2.28i)T + (29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (5.33 + 4.62i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (0.791 + 0.361i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-4.26 + 2.46i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.27 - 0.108i)T + (52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (4.34 + 8.42i)T + (-34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (3.03 - 4.26i)T + (-19.9 - 57.6i)T^{2} \) |
| 67 | \( 1 + (1.08 - 2.70i)T + (-48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (-0.389 - 2.71i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (16.5 + 4.02i)T + (64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (-13.0 - 0.623i)T + (78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (0.139 + 0.160i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (2.36 - 0.225i)T + (87.3 - 16.8i)T^{2} \) |
| 97 | \( 1 + (5.18 - 5.98i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.71429252031662134318309377079, −10.29766302848296871185018534246, −9.104994531469056715545281624098, −8.287555381240718163884536030948, −7.22123923014421718432083193182, −6.37755395751011238548829059546, −5.13256959862423068851864958748, −3.86733850660475617717506817727, −2.93064982662842232326491492750, −1.71237094173996199532528560064,
1.69471299543180408857182980650, 2.96211348349277996462324571615, 4.63235650354247998802422027002, 5.37092090187475562614414394238, 6.15051589536049625882793681383, 7.07903125743611099457821171021, 8.690959338140113275484752839469, 9.427700363356768137687584472982, 9.732637056957332454794934206056, 10.88976946494779322123468686289
