| L(s) = 1 | + 9.48·2-s + 9·3-s + 57.9·4-s + 85.3·6-s − 118.·7-s + 245.·8-s + 81·9-s − 612.·11-s + 521.·12-s + 169·13-s − 1.12e3·14-s + 478.·16-s + 927.·17-s + 768.·18-s + 1.11e3·19-s − 1.06e3·21-s − 5.80e3·22-s − 373.·23-s + 2.21e3·24-s + 1.60e3·26-s + 729·27-s − 6.85e3·28-s − 4.43e3·29-s − 8.19e3·31-s − 3.33e3·32-s − 5.51e3·33-s + 8.79e3·34-s + ⋯ |
| L(s) = 1 | + 1.67·2-s + 0.577·3-s + 1.81·4-s + 0.967·6-s − 0.912·7-s + 1.35·8-s + 0.333·9-s − 1.52·11-s + 1.04·12-s + 0.277·13-s − 1.52·14-s + 0.467·16-s + 0.778·17-s + 0.558·18-s + 0.705·19-s − 0.526·21-s − 2.55·22-s − 0.147·23-s + 0.784·24-s + 0.464·26-s + 0.192·27-s − 1.65·28-s − 0.979·29-s − 1.53·31-s − 0.575·32-s − 0.880·33-s + 1.30·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - 169T \) |
| good | 2 | \( 1 - 9.48T + 32T^{2} \) |
| 7 | \( 1 + 118.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 612.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 927.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.11e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 373.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.43e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.19e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.18e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.01e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.07e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 7.13e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 6.72e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.33e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.06e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.49e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.34e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.13e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.07e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.21e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.14e4T + 8.58e9T^{2} \) |
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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
−8.847942062503719478108054968186, −7.60604456637800196065958990791, −7.13287786505171042571074995643, −5.84648695083420590029345385543, −5.46642461960274584302184171573, −4.34016549294829566938022938578, −3.31905322588906733435974949742, −2.95430473999340418697243901722, −1.80550904378984041155886197547, 0,
1.80550904378984041155886197547, 2.95430473999340418697243901722, 3.31905322588906733435974949742, 4.34016549294829566938022938578, 5.46642461960274584302184171573, 5.84648695083420590029345385543, 7.13287786505171042571074995643, 7.60604456637800196065958990791, 8.847942062503719478108054968186
