| L(s) = 1 | + 2·3-s + 2·5-s − 20·7-s − 3·9-s + 22·11-s + 80·13-s + 4·15-s − 124·17-s − 72·19-s − 40·21-s + 98·23-s − 55·25-s + 34·27-s + 144·29-s + 34·31-s + 44·33-s − 40·35-s + 54·37-s + 160·39-s + 536·41-s + 60·43-s − 6·45-s + 272·47-s − 338·49-s − 248·51-s − 492·53-s + 44·55-s + ⋯ |
| L(s) = 1 | + 0.384·3-s + 0.178·5-s − 1.07·7-s − 1/9·9-s + 0.603·11-s + 1.70·13-s + 0.0688·15-s − 1.76·17-s − 0.869·19-s − 0.415·21-s + 0.888·23-s − 0.439·25-s + 0.242·27-s + 0.922·29-s + 0.196·31-s + 0.232·33-s − 0.193·35-s + 0.239·37-s + 0.656·39-s + 2.04·41-s + 0.212·43-s − 0.0198·45-s + 0.844·47-s − 0.985·49-s − 0.680·51-s − 1.27·53-s + 0.107·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30976 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.211659185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.211659185\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 7 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 59 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 20 T + 738 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 80 T + 4794 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 124 T + 13238 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 72 T + 4214 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 98 T + 22847 T^{2} - 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 144 T + 44554 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 34 T + 57519 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 54 T + 101843 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 536 T + 209618 T^{2} - 536 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 60 T + 159146 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 272 T + 182942 T^{2} - 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 492 T + 348862 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 634 T + 458975 T^{2} + 634 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 840 T + 528794 T^{2} - 840 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 754 T + 742455 T^{2} + 754 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 678 T + 813415 T^{2} - 678 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 400 T + 160962 T^{2} + 400 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 p T - 279966 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 468 T + 1155130 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1842 T + 1935427 T^{2} + 1842 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2194 T + 2966547 T^{2} - 2194 p^{3} T^{3} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.84455765316679115637745619256, −12.00663279820436872040432581770, −11.34193069225278154615372764323, −11.00756263577464032515949169898, −10.62912037527877206940143408445, −9.931930137157842913229251838529, −9.210903034924194971850627559185, −9.113259418582140671562269778803, −8.497232009387628230092352959699, −8.103771142439569495361429473836, −7.07871607976125627098618086045, −6.70633982756460467327052555637, −6.04371483827933391654889792032, −5.96806499701908828217331666444, −4.52369724735107289439189220315, −4.28885625092892669419915841318, −3.35747081780501437802612655775, −2.79897739787279750364358896185, −1.81412119460402935244557463065, −0.67145964999420521334236689867,
0.67145964999420521334236689867, 1.81412119460402935244557463065, 2.79897739787279750364358896185, 3.35747081780501437802612655775, 4.28885625092892669419915841318, 4.52369724735107289439189220315, 5.96806499701908828217331666444, 6.04371483827933391654889792032, 6.70633982756460467327052555637, 7.07871607976125627098618086045, 8.103771142439569495361429473836, 8.497232009387628230092352959699, 9.113259418582140671562269778803, 9.210903034924194971850627559185, 9.931930137157842913229251838529, 10.62912037527877206940143408445, 11.00756263577464032515949169898, 11.34193069225278154615372764323, 12.00663279820436872040432581770, 12.84455765316679115637745619256
