| L(s) = 1 | − 2-s + 3-s + 4-s − 3.32·5-s − 6-s − 3.91·7-s − 8-s + 9-s + 3.32·10-s − 11-s + 12-s − 2.77·13-s + 3.91·14-s − 3.32·15-s + 16-s + 0.807·17-s − 18-s − 7.13·19-s − 3.32·20-s − 3.91·21-s + 22-s − 5.96·23-s − 24-s + 6.04·25-s + 2.77·26-s + 27-s − 3.91·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.48·5-s − 0.408·6-s − 1.47·7-s − 0.353·8-s + 0.333·9-s + 1.05·10-s − 0.301·11-s + 0.288·12-s − 0.769·13-s + 1.04·14-s − 0.858·15-s + 0.250·16-s + 0.195·17-s − 0.235·18-s − 1.63·19-s − 0.743·20-s − 0.854·21-s + 0.213·22-s − 1.24·23-s − 0.204·24-s + 1.20·25-s + 0.544·26-s + 0.192·27-s − 0.739·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2265680081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2265680081\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 5 | \( 1 + 3.32T + 5T^{2} \) |
| 7 | \( 1 + 3.91T + 7T^{2} \) |
| 13 | \( 1 + 2.77T + 13T^{2} \) |
| 17 | \( 1 - 0.807T + 17T^{2} \) |
| 19 | \( 1 + 7.13T + 19T^{2} \) |
| 23 | \( 1 + 5.96T + 23T^{2} \) |
| 29 | \( 1 + 8.37T + 29T^{2} \) |
| 31 | \( 1 + 0.644T + 31T^{2} \) |
| 37 | \( 1 + 8.31T + 37T^{2} \) |
| 41 | \( 1 - 9.01T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + 2.49T + 47T^{2} \) |
| 53 | \( 1 - 0.130T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 - 9.98T + 73T^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 - 3.25T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.283273845552676871696175818111, −7.84350097189377024416199785183, −7.19397709233770115156225638101, −6.55343206257956200279473311520, −5.65660177026168398168460383579, −4.26016256632928677438457130281, −3.78766681361391402568595764003, −2.94003214207033804712280615843, −2.08294961971783177456802230485, −0.27165629202035447610509418346,
0.27165629202035447610509418346, 2.08294961971783177456802230485, 2.94003214207033804712280615843, 3.78766681361391402568595764003, 4.26016256632928677438457130281, 5.65660177026168398168460383579, 6.55343206257956200279473311520, 7.19397709233770115156225638101, 7.84350097189377024416199785183, 8.283273845552676871696175818111
