| L(s) = 1 | + 2-s − 8·3-s − 7·4-s − 5·5-s − 8·6-s + 7·7-s − 15·8-s + 37·9-s − 5·10-s + 12·11-s + 56·12-s − 78·13-s + 7·14-s + 40·15-s + 41·16-s − 94·17-s + 37·18-s + 40·19-s + 35·20-s − 56·21-s + 12·22-s + 32·23-s + 120·24-s + 25·25-s − 78·26-s − 80·27-s − 49·28-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 1.53·3-s − 7/8·4-s − 0.447·5-s − 0.544·6-s + 0.377·7-s − 0.662·8-s + 1.37·9-s − 0.158·10-s + 0.328·11-s + 1.34·12-s − 1.66·13-s + 0.133·14-s + 0.688·15-s + 0.640·16-s − 1.34·17-s + 0.484·18-s + 0.482·19-s + 0.391·20-s − 0.581·21-s + 0.116·22-s + 0.290·23-s + 1.02·24-s + 1/5·25-s − 0.588·26-s − 0.570·27-s − 0.330·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 3 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 94 T + p^{3} T^{2} \) |
| 19 | \( 1 - 40 T + p^{3} T^{2} \) |
| 23 | \( 1 - 32 T + p^{3} T^{2} \) |
| 29 | \( 1 + 50 T + p^{3} T^{2} \) |
| 31 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 37 | \( 1 + 434 T + p^{3} T^{2} \) |
| 41 | \( 1 - 402 T + p^{3} T^{2} \) |
| 43 | \( 1 + 68 T + p^{3} T^{2} \) |
| 47 | \( 1 - 536 T + p^{3} T^{2} \) |
| 53 | \( 1 - 22 T + p^{3} T^{2} \) |
| 59 | \( 1 + 560 T + p^{3} T^{2} \) |
| 61 | \( 1 + 278 T + p^{3} T^{2} \) |
| 67 | \( 1 + 164 T + p^{3} T^{2} \) |
| 71 | \( 1 - 672 T + p^{3} T^{2} \) |
| 73 | \( 1 - 82 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1000 T + p^{3} T^{2} \) |
| 83 | \( 1 + 448 T + p^{3} T^{2} \) |
| 89 | \( 1 + 870 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1026 T + p^{3} T^{2} \) |
| 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
−15.50217950966324923898462215554, −14.26345646777693422232764700602, −12.70453334590557832947232791356, −11.92655234467305331455017611887, −10.74664086894993963800643426567, −9.192146462467410141721051534476, −7.18652248590761618046061851967, −5.45423844932287809114770528907, −4.44336193877597669726946649106, 0,
4.44336193877597669726946649106, 5.45423844932287809114770528907, 7.18652248590761618046061851967, 9.192146462467410141721051534476, 10.74664086894993963800643426567, 11.92655234467305331455017611887, 12.70453334590557832947232791356, 14.26345646777693422232764700602, 15.50217950966324923898462215554
