| L(s) = 1 | − 4.04·2-s − 8.44·3-s − 15.6·4-s − 80.4·5-s + 34.1·6-s + 49·7-s + 192.·8-s − 171.·9-s + 325.·10-s + 132.·12-s − 1.04e3·13-s − 198.·14-s + 679.·15-s − 277.·16-s + 1.37e3·17-s + 693.·18-s − 3.07e3·19-s + 1.25e3·20-s − 413.·21-s − 2.80e3·23-s − 1.62e3·24-s + 3.34e3·25-s + 4.20e3·26-s + 3.50e3·27-s − 767.·28-s + 1.96e3·29-s − 2.74e3·30-s + ⋯ |
| L(s) = 1 | − 0.714·2-s − 0.541·3-s − 0.489·4-s − 1.43·5-s + 0.387·6-s + 0.377·7-s + 1.06·8-s − 0.706·9-s + 1.02·10-s + 0.265·12-s − 1.70·13-s − 0.270·14-s + 0.779·15-s − 0.270·16-s + 1.15·17-s + 0.504·18-s − 1.95·19-s + 0.704·20-s − 0.204·21-s − 1.10·23-s − 0.576·24-s + 1.06·25-s + 1.22·26-s + 0.924·27-s − 0.185·28-s + 0.433·29-s − 0.556·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - 49T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 4.04T + 32T^{2} \) |
| 3 | \( 1 + 8.44T + 243T^{2} \) |
| 5 | \( 1 + 80.4T + 3.12e3T^{2} \) |
| 13 | \( 1 + 1.04e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.37e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 3.07e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.80e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.96e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.66e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.89e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.20e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.03e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 77.6T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.08e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.06e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.49e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.37e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 8.44e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.83e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.02e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.23e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.85e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.41e3T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.786457100604925064397137861965, −8.149029231242603474220433012529, −7.67694032715633811102803567565, −6.63566659785065906394049055117, −5.27863550319407011202207842406, −4.58758510409273492678028783650, −3.72657076377411650174994751876, −2.26688884517417117811447764662, −0.64848329120098837967478892039, 0,
0.64848329120098837967478892039, 2.26688884517417117811447764662, 3.72657076377411650174994751876, 4.58758510409273492678028783650, 5.27863550319407011202207842406, 6.63566659785065906394049055117, 7.67694032715633811102803567565, 8.149029231242603474220433012529, 8.786457100604925064397137861965
