| L(s) = 1 | + 3·2-s + 10·3-s + 4-s + 5·5-s + 30·6-s − 22·7-s − 21·8-s + 73·9-s + 15·10-s − 30·11-s + 10·12-s − 46·13-s − 66·14-s + 50·15-s − 71·16-s + 17·17-s + 219·18-s + 104·19-s + 5·20-s − 220·21-s − 90·22-s + 42·23-s − 210·24-s + 25·25-s − 138·26-s + 460·27-s − 22·28-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 1.92·3-s + 1/8·4-s + 0.447·5-s + 2.04·6-s − 1.18·7-s − 0.928·8-s + 2.70·9-s + 0.474·10-s − 0.822·11-s + 0.240·12-s − 0.981·13-s − 1.25·14-s + 0.860·15-s − 1.10·16-s + 0.242·17-s + 2.86·18-s + 1.25·19-s + 0.0559·20-s − 2.28·21-s − 0.872·22-s + 0.380·23-s − 1.78·24-s + 1/5·25-s − 1.04·26-s + 3.27·27-s − 0.148·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.680256457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.680256457\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - p T \) |
| 17 | \( 1 - p T \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 3 | \( 1 - 10 T + p^{3} T^{2} \) |
| 7 | \( 1 + 22 T + p^{3} T^{2} \) |
| 11 | \( 1 + 30 T + p^{3} T^{2} \) |
| 13 | \( 1 + 46 T + p^{3} T^{2} \) |
| 19 | \( 1 - 104 T + p^{3} T^{2} \) |
| 23 | \( 1 - 42 T + p^{3} T^{2} \) |
| 29 | \( 1 + 66 T + p^{3} T^{2} \) |
| 31 | \( 1 - 194 T + p^{3} T^{2} \) |
| 37 | \( 1 - 206 T + p^{3} T^{2} \) |
| 41 | \( 1 + 126 T + p^{3} T^{2} \) |
| 43 | \( 1 + 388 T + p^{3} T^{2} \) |
| 47 | \( 1 + 540 T + p^{3} T^{2} \) |
| 53 | \( 1 - 78 T + p^{3} T^{2} \) |
| 59 | \( 1 - 432 T + p^{3} T^{2} \) |
| 61 | \( 1 + 10 p T + p^{3} T^{2} \) |
| 67 | \( 1 - 848 T + p^{3} T^{2} \) |
| 71 | \( 1 + 174 T + p^{3} T^{2} \) |
| 73 | \( 1 - 362 T + p^{3} T^{2} \) |
| 79 | \( 1 - 398 T + p^{3} T^{2} \) |
| 83 | \( 1 - 828 T + p^{3} T^{2} \) |
| 89 | \( 1 - 630 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1486 T + p^{3} T^{2} \) |
| 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
−13.56037868544539246007042469245, −13.24574126302381298818062205289, −12.30483613111328331021309934362, −9.835903345403889047959600529799, −9.524493779044116225731876957821, −8.108068285846310080656639825719, −6.78229029383547171406247370707, −4.98146869003944187853035513826, −3.40719408052416579548224842014, −2.66739571901491553444539882875,
2.66739571901491553444539882875, 3.40719408052416579548224842014, 4.98146869003944187853035513826, 6.78229029383547171406247370707, 8.108068285846310080656639825719, 9.524493779044116225731876957821, 9.835903345403889047959600529799, 12.30483613111328331021309934362, 13.24574126302381298818062205289, 13.56037868544539246007042469245
