L(s) = 1 | − 2·2-s − 9·3-s − 28·4-s + 11·5-s + 18·6-s + 120·8-s + 81·9-s − 22·10-s + 269·11-s + 252·12-s − 308·13-s − 99·15-s + 656·16-s + 1.89e3·17-s − 162·18-s − 164·19-s − 308·20-s − 538·22-s − 3.26e3·23-s − 1.08e3·24-s − 3.00e3·25-s + 616·26-s − 729·27-s + 2.41e3·29-s + 198·30-s + 2.84e3·31-s − 5.15e3·32-s + ⋯ |
L(s) = 1 | − 0.353·2-s − 0.577·3-s − 7/8·4-s + 0.196·5-s + 0.204·6-s + 0.662·8-s + 1/3·9-s − 0.0695·10-s + 0.670·11-s + 0.505·12-s − 0.505·13-s − 0.113·15-s + 0.640·16-s + 1.59·17-s − 0.117·18-s − 0.104·19-s − 0.172·20-s − 0.236·22-s − 1.28·23-s − 0.382·24-s − 0.961·25-s + 0.178·26-s − 0.192·27-s + 0.533·29-s + 0.0401·30-s + 0.530·31-s − 0.889·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{2} T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + p T + p^{5} T^{2} \) |
| 5 | \( 1 - 11 T + p^{5} T^{2} \) |
| 11 | \( 1 - 269 T + p^{5} T^{2} \) |
| 13 | \( 1 + 308 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1896 T + p^{5} T^{2} \) |
| 19 | \( 1 + 164 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3264 T + p^{5} T^{2} \) |
| 29 | \( 1 - 2417 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2841 T + p^{5} T^{2} \) |
| 37 | \( 1 + 11328 T + p^{5} T^{2} \) |
| 41 | \( 1 + 16856 T + p^{5} T^{2} \) |
| 43 | \( 1 + 7894 T + p^{5} T^{2} \) |
| 47 | \( 1 - 21102 T + p^{5} T^{2} \) |
| 53 | \( 1 + 29691 T + p^{5} T^{2} \) |
| 59 | \( 1 + 8163 T + p^{5} T^{2} \) |
| 61 | \( 1 - 15166 T + p^{5} T^{2} \) |
| 67 | \( 1 + 32078 T + p^{5} T^{2} \) |
| 71 | \( 1 + 38274 T + p^{5} T^{2} \) |
| 73 | \( 1 - 34866 T + p^{5} T^{2} \) |
| 79 | \( 1 - 13529 T + p^{5} T^{2} \) |
| 83 | \( 1 + 68103 T + p^{5} T^{2} \) |
| 89 | \( 1 + 114922 T + p^{5} T^{2} \) |
| 97 | \( 1 - 154959 T + p^{5} T^{2} \) |
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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
−11.81866921001519988746856245873, −10.22170792984889296954140196769, −9.828907223110264927121022561691, −8.543177978043506276008379336420, −7.48756535656876164722452100236, −6.04049124479231519485094262854, −4.96330654375456483913593837991, −3.70209687220315979017336671593, −1.45625033814366614583044636558, 0,
1.45625033814366614583044636558, 3.70209687220315979017336671593, 4.96330654375456483913593837991, 6.04049124479231519485094262854, 7.48756535656876164722452100236, 8.543177978043506276008379336420, 9.828907223110264927121022561691, 10.22170792984889296954140196769, 11.81866921001519988746856245873