| L(s) = 1 | + 5·3-s − 5·5-s + 30·7-s − 27·9-s + 71·11-s − 35·13-s − 25·15-s + 38·19-s + 150·21-s + 5·23-s − 25·25-s − 260·27-s + 155·29-s + 88·31-s + 355·33-s − 150·35-s + 380·37-s − 175·39-s − 142·41-s − 155·43-s + 135·45-s + 455·47-s + 121·49-s − 275·53-s − 355·55-s + 190·57-s + 873·59-s + ⋯ |
| L(s) = 1 | + 0.962·3-s − 0.447·5-s + 1.61·7-s − 9-s + 1.94·11-s − 0.746·13-s − 0.430·15-s + 0.458·19-s + 1.55·21-s + 0.0453·23-s − 1/5·25-s − 1.85·27-s + 0.992·29-s + 0.509·31-s + 1.87·33-s − 0.724·35-s + 1.68·37-s − 0.718·39-s − 0.540·41-s − 0.549·43-s + 0.447·45-s + 1.41·47-s + 0.352·49-s − 0.712·53-s − 0.870·55-s + 0.441·57-s + 1.92·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.238788859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.238788859\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 5 T + 52 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + p T + 2 p^{2} T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 30 T + 779 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 71 T + 3716 T^{2} - 71 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 35 T + 3702 T^{2} + 35 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 9529 T^{2} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 5 T - 4378 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 155 T + 19928 T^{2} - 155 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 88 T + 8718 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 380 T + 136878 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 142 T + 135458 T^{2} + 142 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 155 T + 52086 T^{2} + 155 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 455 T + 255038 T^{2} - 455 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 275 T + 191846 T^{2} + 275 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 873 T + 591184 T^{2} - 873 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 445 T + 250812 T^{2} - 445 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 645 T + 674834 T^{2} + 645 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1712 T + 1445258 T^{2} - 1712 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 990 T + 989267 T^{2} + 990 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1274 T + 1391022 T^{2} - 1274 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 90 T + 1038382 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 888 T + 1554274 T^{2} + 888 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 710 T + 220818 T^{2} - 710 p^{3} T^{3} + p^{6} T^{4} \) |
| 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
−11.44074591378052331103512113893, −11.39894769220175226730446579456, −10.69560153202344486103106607243, −9.987638382394648266856100206738, −9.453970025955761482759052546352, −9.116095452896825824829732250907, −8.443576186931964498924018022786, −8.409857930690628276882263752098, −7.72315221331815903144200362301, −7.50924800718428842538387912154, −6.49576162412878367107996882319, −6.35891654005827199974563264306, −5.20387394885263760812976044008, −5.11207799281862697049910370706, −3.99741993822811759751474734668, −3.99665700438603626216575014331, −2.93116964125156225909801857175, −2.41965971956718394396818241026, −1.56850168219061567845452618242, −0.76081911945042079563190465052,
0.76081911945042079563190465052, 1.56850168219061567845452618242, 2.41965971956718394396818241026, 2.93116964125156225909801857175, 3.99665700438603626216575014331, 3.99741993822811759751474734668, 5.11207799281862697049910370706, 5.20387394885263760812976044008, 6.35891654005827199974563264306, 6.49576162412878367107996882319, 7.50924800718428842538387912154, 7.72315221331815903144200362301, 8.409857930690628276882263752098, 8.443576186931964498924018022786, 9.116095452896825824829732250907, 9.453970025955761482759052546352, 9.987638382394648266856100206738, 10.69560153202344486103106607243, 11.39894769220175226730446579456, 11.44074591378052331103512113893
