| L(s) = 1 | + 4·2-s + 9.42·3-s + 16·4-s − 32.1·5-s + 37.7·6-s − 9.66·7-s + 64·8-s − 154.·9-s − 128.·10-s − 114.·11-s + 150.·12-s + 37.2·13-s − 38.6·14-s − 303.·15-s + 256·16-s + 946.·17-s − 616.·18-s + 571.·19-s − 515.·20-s − 91.0·21-s − 456.·22-s + 3.65e3·23-s + 603.·24-s − 2.08e3·25-s + 149.·26-s − 3.74e3·27-s − 154.·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.604·3-s + 0.5·4-s − 0.575·5-s + 0.427·6-s − 0.0745·7-s + 0.353·8-s − 0.634·9-s − 0.407·10-s − 0.284·11-s + 0.302·12-s + 0.0611·13-s − 0.0527·14-s − 0.348·15-s + 0.250·16-s + 0.794·17-s − 0.448·18-s + 0.363·19-s − 0.287·20-s − 0.0450·21-s − 0.200·22-s + 1.43·23-s + 0.213·24-s − 0.668·25-s + 0.0432·26-s − 0.988·27-s − 0.0372·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 431 | \( 1 - 1.85e5T \) |
| good | 3 | \( 1 - 9.42T + 243T^{2} \) |
| 5 | \( 1 + 32.1T + 3.12e3T^{2} \) |
| 7 | \( 1 + 9.66T + 1.68e4T^{2} \) |
| 11 | \( 1 + 114.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 37.2T + 3.71e5T^{2} \) |
| 17 | \( 1 - 946.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 571.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.65e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 524.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.16e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.75e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.31e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.59e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.66e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.17e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.31e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.33e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.48e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.49e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.93e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.27e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.15e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.37e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.35e4T + 8.58e9T^{2} \) |
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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
−8.965974516843017867953998968326, −7.906429892168354223570944313067, −7.50696904023124810799708552128, −6.27037586668296511800534912142, −5.41309587768272598331096418969, −4.44506960004953186829885790375, −3.32503827788290774643248752905, −2.87183411770027915770535271032, −1.48102391154436066599162504340, 0,
1.48102391154436066599162504340, 2.87183411770027915770535271032, 3.32503827788290774643248752905, 4.44506960004953186829885790375, 5.41309587768272598331096418969, 6.27037586668296511800534912142, 7.50696904023124810799708552128, 7.906429892168354223570944313067, 8.965974516843017867953998968326
