| L(s) = 1 | − 2·3-s − 5-s + 7-s + 9-s − 2·11-s + 2·15-s + 6·17-s − 6·19-s − 2·21-s + 6·23-s + 25-s + 4·27-s + 6·29-s + 6·31-s + 4·33-s − 35-s + 6·37-s + 10·41-s + 6·43-s − 45-s + 49-s − 12·51-s − 6·53-s + 2·55-s + 12·57-s − 10·59-s − 2·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.516·15-s + 1.45·17-s − 1.37·19-s − 0.436·21-s + 1.25·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s + 1.07·31-s + 0.696·33-s − 0.169·35-s + 0.986·37-s + 1.56·41-s + 0.914·43-s − 0.149·45-s + 1/7·49-s − 1.68·51-s − 0.824·53-s + 0.269·55-s + 1.58·57-s − 1.30·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.084933046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.084933046\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.38182166553240, −12.11591949694781, −11.52658803539761, −10.98834468675849, −10.87679210190896, −10.44108361888210, −9.904243941621624, −9.342819518043096, −8.851609222133533, −8.185030248633611, −7.905229995522238, −7.571444037868464, −6.749602382870974, −6.453541851714907, −5.998022622854349, −5.468686424033955, −4.976045556861618, −4.626215796487772, −4.165818849101759, −3.444489420234348, −2.739658058099180, −2.512861795007310, −1.480169834371493, −0.7735644209932395, −0.5960766309539369,
0.5960766309539369, 0.7735644209932395, 1.480169834371493, 2.512861795007310, 2.739658058099180, 3.444489420234348, 4.165818849101759, 4.626215796487772, 4.976045556861618, 5.468686424033955, 5.998022622854349, 6.453541851714907, 6.749602382870974, 7.571444037868464, 7.905229995522238, 8.185030248633611, 8.851609222133533, 9.342819518043096, 9.904243941621624, 10.44108361888210, 10.87679210190896, 10.98834468675849, 11.52658803539761, 12.11591949694781, 12.38182166553240
