| L(s) = 1 | − 2.63·2-s + 28.9·3-s − 25.0·4-s − 25·5-s − 76.3·6-s − 13.2·7-s + 150.·8-s + 595.·9-s + 65.9·10-s − 121·11-s − 725.·12-s + 1.11e3·13-s + 34.9·14-s − 724.·15-s + 404.·16-s − 421.·17-s − 1.57e3·18-s + 361·19-s + 626.·20-s − 383.·21-s + 319.·22-s + 60.1·23-s + 4.35e3·24-s + 625·25-s − 2.94e3·26-s + 1.02e4·27-s + 331.·28-s + ⋯ |
| L(s) = 1 | − 0.466·2-s + 1.85·3-s − 0.782·4-s − 0.447·5-s − 0.866·6-s − 0.102·7-s + 0.831·8-s + 2.45·9-s + 0.208·10-s − 0.301·11-s − 1.45·12-s + 1.83·13-s + 0.0476·14-s − 0.830·15-s + 0.395·16-s − 0.353·17-s − 1.14·18-s + 0.229·19-s + 0.349·20-s − 0.189·21-s + 0.140·22-s + 0.0237·23-s + 1.54·24-s + 0.200·25-s − 0.854·26-s + 2.69·27-s + 0.0799·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.206156551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.206156551\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 + 2.63T + 32T^{2} \) |
| 3 | \( 1 - 28.9T + 243T^{2} \) |
| 7 | \( 1 + 13.2T + 1.68e4T^{2} \) |
| 13 | \( 1 - 1.11e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 421.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 60.1T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.78e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.23e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.73e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.45e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.35e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.03e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.53e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.88e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.98e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.52e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.34e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.40e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.27e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.11e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.25e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.00e5T + 8.58e9T^{2} \) |
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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
−8.990266771844548977284716245462, −8.522016344636053588866830753214, −7.82672536871270566377660980943, −7.21782927900963328237987853121, −5.80389736782926796449629396944, −4.31555761490389193920334554130, −3.87012137084865997553035094815, −2.99168607602894920587464077324, −1.76268354382011779957743024456, −0.803671475967798911230307197822,
0.803671475967798911230307197822, 1.76268354382011779957743024456, 2.99168607602894920587464077324, 3.87012137084865997553035094815, 4.31555761490389193920334554130, 5.80389736782926796449629396944, 7.21782927900963328237987853121, 7.82672536871270566377660980943, 8.522016344636053588866830753214, 8.990266771844548977284716245462
