| L(s) = 1 | − 2-s − 3-s + 4-s − 2.78·5-s + 6-s + 3.42·7-s − 8-s + 9-s + 2.78·10-s − 6.40·11-s − 12-s − 1.09·13-s − 3.42·14-s + 2.78·15-s + 16-s + 17-s − 18-s + 6.34·19-s − 2.78·20-s − 3.42·21-s + 6.40·22-s − 0.0945·23-s + 24-s + 2.73·25-s + 1.09·26-s − 27-s + 3.42·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.24·5-s + 0.408·6-s + 1.29·7-s − 0.353·8-s + 0.333·9-s + 0.879·10-s − 1.93·11-s − 0.288·12-s − 0.304·13-s − 0.914·14-s + 0.717·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 1.45·19-s − 0.621·20-s − 0.746·21-s + 1.36·22-s − 0.0197·23-s + 0.204·24-s + 0.546·25-s + 0.215·26-s − 0.192·27-s + 0.646·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5840373233\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5840373233\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 + 2.78T + 5T^{2} \) |
| 7 | \( 1 - 3.42T + 7T^{2} \) |
| 11 | \( 1 + 6.40T + 11T^{2} \) |
| 13 | \( 1 + 1.09T + 13T^{2} \) |
| 19 | \( 1 - 6.34T + 19T^{2} \) |
| 23 | \( 1 + 0.0945T + 23T^{2} \) |
| 29 | \( 1 + 3.02T + 29T^{2} \) |
| 31 | \( 1 + 2.51T + 31T^{2} \) |
| 37 | \( 1 + 4.03T + 37T^{2} \) |
| 41 | \( 1 - 3.94T + 41T^{2} \) |
| 43 | \( 1 + 4.15T + 43T^{2} \) |
| 47 | \( 1 - 3.44T + 47T^{2} \) |
| 53 | \( 1 - 0.825T + 53T^{2} \) |
| 61 | \( 1 + 1.88T + 61T^{2} \) |
| 67 | \( 1 + 6.99T + 67T^{2} \) |
| 71 | \( 1 + 8.64T + 71T^{2} \) |
| 73 | \( 1 - 6.87T + 73T^{2} \) |
| 79 | \( 1 - 6.85T + 79T^{2} \) |
| 83 | \( 1 + 0.152T + 83T^{2} \) |
| 89 | \( 1 + 9.59T + 89T^{2} \) |
| 97 | \( 1 - 1.52T + 97T^{2} \) |
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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
−7.892290212953599938811256894847, −7.56291815228671976474870813905, −7.13914198669834966653640653941, −5.77006618484969342640518732772, −5.20717770693221727688128326048, −4.65738886514840611479224304548, −3.60154748470136063416633826422, −2.68826901446299371154547603468, −1.61152737186255826439554059865, −0.46508801446345660511828806478,
0.46508801446345660511828806478, 1.61152737186255826439554059865, 2.68826901446299371154547603468, 3.60154748470136063416633826422, 4.65738886514840611479224304548, 5.20717770693221727688128326048, 5.77006618484969342640518732772, 7.13914198669834966653640653941, 7.56291815228671976474870813905, 7.892290212953599938811256894847
