| L(s) = 1 | − 2-s + 3-s + 4-s − 4.40·5-s − 6-s − 0.551·7-s − 8-s + 9-s + 4.40·10-s + 1.47·11-s + 12-s − 2.62·13-s + 0.551·14-s − 4.40·15-s + 16-s − 1.48·17-s − 18-s + 2.46·19-s − 4.40·20-s − 0.551·21-s − 1.47·22-s − 23-s − 24-s + 14.4·25-s + 2.62·26-s + 27-s − 0.551·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.97·5-s − 0.408·6-s − 0.208·7-s − 0.353·8-s + 0.333·9-s + 1.39·10-s + 0.444·11-s + 0.288·12-s − 0.727·13-s + 0.147·14-s − 1.13·15-s + 0.250·16-s − 0.360·17-s − 0.235·18-s + 0.566·19-s − 0.986·20-s − 0.120·21-s − 0.314·22-s − 0.208·23-s − 0.204·24-s + 2.88·25-s + 0.514·26-s + 0.192·27-s − 0.104·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + 4.40T + 5T^{2} \) |
| 7 | \( 1 + 0.551T + 7T^{2} \) |
| 11 | \( 1 - 1.47T + 11T^{2} \) |
| 13 | \( 1 + 2.62T + 13T^{2} \) |
| 17 | \( 1 + 1.48T + 17T^{2} \) |
| 19 | \( 1 - 2.46T + 19T^{2} \) |
| 31 | \( 1 - 5.87T + 31T^{2} \) |
| 37 | \( 1 - 4.22T + 37T^{2} \) |
| 41 | \( 1 + 0.176T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 + 3.90T + 53T^{2} \) |
| 59 | \( 1 - 9.14T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 - 6.61T + 73T^{2} \) |
| 79 | \( 1 + 1.53T + 79T^{2} \) |
| 83 | \( 1 + 5.66T + 83T^{2} \) |
| 89 | \( 1 + 6.00T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−8.033425280244344871730167380733, −7.53429302683910003198798419067, −7.03064216557184640527348051845, −6.15614789561547454589583403927, −4.73397804536037602947924394714, −4.20492927554533029888724695857, −3.27663752488207807709485966493, −2.65462199985882722696557368439, −1.15136100559167426286202914305, 0,
1.15136100559167426286202914305, 2.65462199985882722696557368439, 3.27663752488207807709485966493, 4.20492927554533029888724695857, 4.73397804536037602947924394714, 6.15614789561547454589583403927, 7.03064216557184640527348051845, 7.53429302683910003198798419067, 8.033425280244344871730167380733
