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 | \( 1 + (-12.5 - 14.5i)T + (-13.8 + 96.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.99272861654331624443098301666, −10.48734176101784710406534434673, −8.964199080543716806297768967319, −8.293696595255821807035831281493, −7.65774054076391342086111169230, −6.30321560401081566546127242416, −5.27705102279980533895820179770, −4.14077235619189731507449880533, −2.59488077414473819940160758273, −0.988385084609548904694128196553,
0.63297872673905837138009208627, 3.31116139570883067782970727815, 4.37815843435914536252759144913, 5.12242311624978472171565045438, 6.85366874620423112348197686298, 7.55107581308443671534211080155, 8.184348199818533443577188286463, 8.909643134482878421330917863460, 10.22649220830689967825753574425, 10.92462365927715626560464295986