| L(s) = 1 | + (−0.945 − 1.16i)2-s + (−0.790 + 0.684i)3-s + (−0.0513 + 0.245i)4-s + (−0.107 − 0.340i)5-s + (1.54 + 0.273i)6-s + (0.296 − 0.861i)7-s + (−2.33 + 1.20i)8-s + (−0.273 + 1.88i)9-s + (−0.294 + 0.447i)10-s + (2.44 − 0.789i)11-s + (−0.127 − 0.229i)12-s + (−5.63 + 1.90i)13-s + (−1.28 + 0.469i)14-s + (0.318 + 0.195i)15-s + (4.06 + 1.78i)16-s + (−3.07 − 1.53i)17-s + ⋯ |
| L(s) = 1 | + (−0.668 − 0.823i)2-s + (−0.456 + 0.395i)3-s + (−0.0256 + 0.122i)4-s + (−0.0481 − 0.152i)5-s + (0.630 + 0.111i)6-s + (0.112 − 0.325i)7-s + (−0.824 + 0.424i)8-s + (−0.0911 + 0.629i)9-s + (−0.0930 + 0.141i)10-s + (0.735 − 0.238i)11-s + (−0.0367 − 0.0661i)12-s + (−1.56 + 0.527i)13-s + (−0.343 + 0.125i)14-s + (0.0821 + 0.0504i)15-s + (1.01 + 0.445i)16-s + (−0.746 − 0.372i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.274 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.289014 + 0.218053i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.289014 + 0.218053i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 503 | \( 1 + (-12.8 + 18.3i)T \) |
| good | 2 | \( 1 + (0.945 + 1.16i)T + (-0.410 + 1.95i)T^{2} \) |
| 3 | \( 1 + (0.790 - 0.684i)T + (0.430 - 2.96i)T^{2} \) |
| 5 | \( 1 + (0.107 + 0.340i)T + (-4.08 + 2.87i)T^{2} \) |
| 7 | \( 1 + (-0.296 + 0.861i)T + (-5.51 - 4.31i)T^{2} \) |
| 11 | \( 1 + (-2.44 + 0.789i)T + (8.91 - 6.44i)T^{2} \) |
| 13 | \( 1 + (5.63 - 1.90i)T + (10.3 - 7.87i)T^{2} \) |
| 17 | \( 1 + (3.07 + 1.53i)T + (10.2 + 13.5i)T^{2} \) |
| 19 | \( 1 + (0.257 + 0.578i)T + (-12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (2.21 - 7.66i)T + (-19.4 - 12.2i)T^{2} \) |
| 29 | \( 1 + (0.698 - 5.84i)T + (-28.1 - 6.83i)T^{2} \) |
| 31 | \( 1 + (-0.275 - 3.37i)T + (-30.5 + 5.02i)T^{2} \) |
| 37 | \( 1 + (-0.501 - 0.633i)T + (-8.49 + 36.0i)T^{2} \) |
| 41 | \( 1 + (2.08 - 10.6i)T + (-37.9 - 15.5i)T^{2} \) |
| 43 | \( 1 + (0.678 - 3.69i)T + (-40.1 - 15.2i)T^{2} \) |
| 47 | \( 1 + (3.27 - 0.123i)T + (46.8 - 3.52i)T^{2} \) |
| 53 | \( 1 + (-1.71 + 3.85i)T + (-35.4 - 39.4i)T^{2} \) |
| 59 | \( 1 + (3.88 - 9.34i)T + (-41.5 - 41.8i)T^{2} \) |
| 61 | \( 1 + (0.738 + 1.21i)T + (-28.2 + 54.0i)T^{2} \) |
| 67 | \( 1 + (2.15 - 1.32i)T + (30.3 - 59.7i)T^{2} \) |
| 71 | \( 1 + (1.46 + 9.30i)T + (-67.5 + 21.8i)T^{2} \) |
| 73 | \( 1 + (0.798 + 0.764i)T + (3.19 + 72.9i)T^{2} \) |
| 79 | \( 1 + (0.297 + 5.27i)T + (-78.4 + 8.88i)T^{2} \) |
| 83 | \( 1 + (-6.45 - 12.3i)T + (-47.3 + 68.1i)T^{2} \) |
| 89 | \( 1 + (3.50 + 13.3i)T + (-77.5 + 43.6i)T^{2} \) |
| 97 | \( 1 + (3.44 + 14.6i)T + (-86.7 + 43.3i)T^{2} \) |
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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
−11.09687453223019462073241746706, −10.22792425739438908698469962877, −9.552604921340986948421872475113, −8.805054571379652583590472919086, −7.62436471507779243702542298194, −6.53509455448604084806988828409, −5.27399000263174462412426458506, −4.47423833458573209753292915518, −2.88100065678341592089237072098, −1.57490388797376566937924040593,
0.27930875113453324344557627900, 2.47048897057164872110675413326, 4.00393160744418326192296192842, 5.45427003243735626040479307294, 6.49228297819257640068761334926, 6.97211229477393631194658806328, 7.938455586025160740331005853701, 8.879941821978495683743947945293, 9.569828226121356473315698770257, 10.60834298225483459736556507564
