| L(s) = 1 | + (1.54 − 1.21i)2-s + (−0.814 + 0.580i)3-s + (0.436 − 1.80i)4-s + (−4.04 − 0.779i)5-s + (−0.552 + 1.88i)6-s + (0.0404 + 2.64i)7-s + (0.120 + 0.263i)8-s + (0.327 − 0.945i)9-s + (−7.18 + 3.70i)10-s + (−3.14 + 3.99i)11-s + (0.688 + 1.71i)12-s + (0.309 − 0.481i)13-s + (3.27 + 4.03i)14-s + (3.74 − 1.71i)15-s + (3.79 + 1.95i)16-s + (−2.05 + 1.95i)17-s + ⋯ |
| L(s) = 1 | + (1.09 − 0.857i)2-s + (−0.470 + 0.334i)3-s + (0.218 − 0.900i)4-s + (−1.80 − 0.348i)5-s + (−0.225 + 0.768i)6-s + (0.0152 + 0.999i)7-s + (0.0425 + 0.0931i)8-s + (0.109 − 0.315i)9-s + (−2.27 + 1.17i)10-s + (−0.948 + 1.20i)11-s + (0.198 + 0.496i)12-s + (0.0857 − 0.133i)13-s + (0.874 + 1.07i)14-s + (0.967 − 0.441i)15-s + (0.949 + 0.489i)16-s + (−0.497 + 0.474i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0447 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0447 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.528560 + 0.552743i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.528560 + 0.552743i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.0404 - 2.64i)T \) |
| 23 | \( 1 + (4.78 - 0.380i)T \) |
| good | 2 | \( 1 + (-1.54 + 1.21i)T + (0.471 - 1.94i)T^{2} \) |
| 5 | \( 1 + (4.04 + 0.779i)T + (4.64 + 1.85i)T^{2} \) |
| 11 | \( 1 + (3.14 - 3.99i)T + (-2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-0.309 + 0.481i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (2.05 - 1.95i)T + (0.808 - 16.9i)T^{2} \) |
| 19 | \( 1 + (3.00 + 2.86i)T + (0.904 + 18.9i)T^{2} \) |
| 29 | \( 1 + (1.31 + 0.386i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.329 + 3.44i)T + (-30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (-7.06 - 2.44i)T + (29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (-5.88 - 5.09i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (10.9 + 4.98i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (10.5 - 6.11i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.07 - 0.0511i)T + (52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (0.353 + 0.685i)T + (-34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (2.47 - 3.48i)T + (-19.9 - 57.6i)T^{2} \) |
| 67 | \( 1 + (-3.80 + 9.49i)T + (-48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (0.213 + 1.48i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-8.43 - 2.04i)T + (64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (0.678 + 0.0323i)T + (78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (-5.76 - 6.65i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-16.5 + 1.58i)T + (87.3 - 16.8i)T^{2} \) |
| 97 | \( 1 + (9.64 - 11.1i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.41337978074179032190240162290, −10.87149500314238296990117208360, −9.704081480281930888898409765529, −8.351109849214660505787527967616, −7.76314776129614763070662522548, −6.26660113211752459430212560459, −4.90025217818399818609389833082, −4.57091953415840946358536411764, −3.53056262746202864865413558257, −2.27981849203600557054325996579,
0.33532330234799028958652958686, 3.31265359114994871329202190592, 4.08810194302484669845814391797, 4.94728006518665483268898911700, 6.19895371157103009291791890150, 6.95291356109054785514450505468, 7.80796633342354395522063501810, 8.236393157405680628437790931206, 10.25600329011914439635608455993, 11.03491508537525026577255043879
