| L(s) = 1 | + (−0.929 + 0.730i)2-s + (−0.814 + 0.580i)3-s + (−0.142 + 0.585i)4-s + (−3.37 − 0.650i)5-s + (0.333 − 1.13i)6-s + (2.36 − 1.17i)7-s + (−1.27 − 2.79i)8-s + (0.327 − 0.945i)9-s + (3.60 − 1.86i)10-s + (−2.39 + 3.04i)11-s + (−0.224 − 0.559i)12-s + (1.88 − 2.93i)13-s + (−1.34 + 2.82i)14-s + (3.12 − 1.42i)15-s + (2.16 + 1.11i)16-s + (1.19 − 1.14i)17-s + ⋯ |
| L(s) = 1 | + (−0.656 + 0.516i)2-s + (−0.470 + 0.334i)3-s + (−0.0710 + 0.292i)4-s + (−1.50 − 0.290i)5-s + (0.135 − 0.463i)6-s + (0.895 − 0.444i)7-s + (−0.451 − 0.989i)8-s + (0.109 − 0.315i)9-s + (1.14 − 0.588i)10-s + (−0.720 + 0.916i)11-s + (−0.0646 − 0.161i)12-s + (0.522 − 0.813i)13-s + (−0.358 + 0.754i)14-s + (0.806 − 0.368i)15-s + (0.540 + 0.278i)16-s + (0.290 − 0.277i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.562964 + 0.0699111i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.562964 + 0.0699111i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.36 + 1.17i)T \) |
| 23 | \( 1 + (-4.73 - 0.781i)T \) |
| good | 2 | \( 1 + (0.929 - 0.730i)T + (0.471 - 1.94i)T^{2} \) |
| 5 | \( 1 + (3.37 + 0.650i)T + (4.64 + 1.85i)T^{2} \) |
| 11 | \( 1 + (2.39 - 3.04i)T + (-2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-1.88 + 2.93i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-1.19 + 1.14i)T + (0.808 - 16.9i)T^{2} \) |
| 19 | \( 1 + (1.99 + 1.90i)T + (0.904 + 18.9i)T^{2} \) |
| 29 | \( 1 + (-4.78 - 1.40i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (0.359 - 3.76i)T + (-30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (-2.75 - 0.952i)T + (29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (0.0864 + 0.0749i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (3.62 + 1.65i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-7.77 + 4.49i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.77 + 0.179i)T + (52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (2.01 + 3.90i)T + (-34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (-6.69 + 9.39i)T + (-19.9 - 57.6i)T^{2} \) |
| 67 | \( 1 + (3.13 - 7.83i)T + (-48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (2.25 + 15.6i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (0.107 + 0.0260i)T + (64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (-6.52 - 0.310i)T + (78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (1.35 + 1.56i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-13.3 + 1.27i)T + (87.3 - 16.8i)T^{2} \) |
| 97 | \( 1 + (-12.5 + 14.5i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.92462365927715626560464295986, −10.22649220830689967825753574425, −8.909643134482878421330917863460, −8.184348199818533443577188286463, −7.55107581308443671534211080155, −6.85366874620423112348197686298, −5.12242311624978472171565045438, −4.37815843435914536252759144913, −3.31116139570883067782970727815, −0.63297872673905837138009208627,
0.988385084609548904694128196553, 2.59488077414473819940160758273, 4.14077235619189731507449880533, 5.27705102279980533895820179770, 6.30321560401081566546127242416, 7.65774054076391342086111169230, 8.293696595255821807035831281493, 8.964199080543716806297768967319, 10.48734176101784710406534434673, 10.99272861654331624443098301666
