| L(s) = 1 | + (−0.225 − 0.277i)2-s + (−0.186 + 0.161i)3-s + (0.383 − 1.83i)4-s + (0.927 + 2.92i)5-s + (0.0866 + 0.0153i)6-s + (−0.576 + 1.67i)7-s + (−1.22 + 0.632i)8-s + (−0.421 + 2.90i)9-s + (0.602 − 0.916i)10-s + (−3.22 + 1.04i)11-s + (0.224 + 0.403i)12-s + (−2.34 + 0.792i)13-s + (0.593 − 0.216i)14-s + (−0.645 − 0.396i)15-s + (−2.97 − 1.30i)16-s + (−5.70 − 2.84i)17-s + ⋯ |
| L(s) = 1 | + (−0.159 − 0.195i)2-s + (−0.107 + 0.0931i)3-s + (0.191 − 0.916i)4-s + (0.414 + 1.30i)5-s + (0.0353 + 0.00626i)6-s + (−0.217 + 0.632i)7-s + (−0.434 + 0.223i)8-s + (−0.140 + 0.969i)9-s + (0.190 − 0.289i)10-s + (−0.971 + 0.314i)11-s + (0.0647 + 0.116i)12-s + (−0.651 + 0.219i)13-s + (0.158 − 0.0579i)14-s + (−0.166 − 0.102i)15-s + (−0.744 − 0.326i)16-s + (−1.38 − 0.690i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.201 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.562704 + 0.690153i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.562704 + 0.690153i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 503 | \( 1 + (-2.93 + 22.2i)T \) |
| good | 2 | \( 1 + (0.225 + 0.277i)T + (-0.410 + 1.95i)T^{2} \) |
| 3 | \( 1 + (0.186 - 0.161i)T + (0.430 - 2.96i)T^{2} \) |
| 5 | \( 1 + (-0.927 - 2.92i)T + (-4.08 + 2.87i)T^{2} \) |
| 7 | \( 1 + (0.576 - 1.67i)T + (-5.51 - 4.31i)T^{2} \) |
| 11 | \( 1 + (3.22 - 1.04i)T + (8.91 - 6.44i)T^{2} \) |
| 13 | \( 1 + (2.34 - 0.792i)T + (10.3 - 7.87i)T^{2} \) |
| 17 | \( 1 + (5.70 + 2.84i)T + (10.2 + 13.5i)T^{2} \) |
| 19 | \( 1 + (-2.32 - 5.21i)T + (-12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (-0.793 + 2.74i)T + (-19.4 - 12.2i)T^{2} \) |
| 29 | \( 1 + (-0.610 + 5.10i)T + (-28.1 - 6.83i)T^{2} \) |
| 31 | \( 1 + (-0.577 - 7.07i)T + (-30.5 + 5.02i)T^{2} \) |
| 37 | \( 1 + (-6.67 - 8.43i)T + (-8.49 + 36.0i)T^{2} \) |
| 41 | \( 1 + (-1.21 + 6.19i)T + (-37.9 - 15.5i)T^{2} \) |
| 43 | \( 1 + (0.335 - 1.82i)T + (-40.1 - 15.2i)T^{2} \) |
| 47 | \( 1 + (-3.35 + 0.126i)T + (46.8 - 3.52i)T^{2} \) |
| 53 | \( 1 + (-3.31 + 7.43i)T + (-35.4 - 39.4i)T^{2} \) |
| 59 | \( 1 + (1.19 - 2.86i)T + (-41.5 - 41.8i)T^{2} \) |
| 61 | \( 1 + (-0.0577 - 0.0953i)T + (-28.2 + 54.0i)T^{2} \) |
| 67 | \( 1 + (-8.58 + 5.27i)T + (30.3 - 59.7i)T^{2} \) |
| 71 | \( 1 + (-2.56 - 16.2i)T + (-67.5 + 21.8i)T^{2} \) |
| 73 | \( 1 + (-2.28 - 2.18i)T + (3.19 + 72.9i)T^{2} \) |
| 79 | \( 1 + (-0.794 - 14.0i)T + (-78.4 + 8.88i)T^{2} \) |
| 83 | \( 1 + (2.37 + 4.54i)T + (-47.3 + 68.1i)T^{2} \) |
| 89 | \( 1 + (2.02 + 7.71i)T + (-77.5 + 43.6i)T^{2} \) |
| 97 | \( 1 + (4.15 + 17.6i)T + (-86.7 + 43.3i)T^{2} \) |
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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
−11.03518880339043809818403518017, −10.14190844648202227822858381567, −9.919138800145433492282061926188, −8.609027144415882362601397197935, −7.37221145797957106382225068600, −6.53278074671998896551963604230, −5.62311292834801897705403682035, −4.76475768863376828477314004190, −2.64788545707821124401370220622, −2.28050200887682204495206590081,
0.52499934072041497636083153126, 2.54959149456828270951800580287, 3.90279304985729318645640878663, 4.93563089952352751008109246790, 6.09839420981388840991871593190, 7.14415408342457571787062498018, 7.963945066474379255530504142407, 9.044624876195334779663696375382, 9.339522037503036606634458022086, 10.75618242786518963834721688177
