| L(s) = 1 | + 2.50·3-s + 0.533·5-s − 0.354·7-s + 3.29·9-s − 4.18·11-s − 3.72·13-s + 1.33·15-s − 6.46·17-s + 1.31·19-s − 0.890·21-s + 4.10·23-s − 4.71·25-s + 0.728·27-s − 8.85·29-s − 5.11·31-s − 10.5·33-s − 0.189·35-s + 5.41·37-s − 9.34·39-s + 11.8·41-s − 0.673·43-s + 1.75·45-s − 5.22·47-s − 6.87·49-s − 16.2·51-s − 9.92·53-s − 2.23·55-s + ⋯ |
| L(s) = 1 | + 1.44·3-s + 0.238·5-s − 0.134·7-s + 1.09·9-s − 1.26·11-s − 1.03·13-s + 0.345·15-s − 1.56·17-s + 0.300·19-s − 0.194·21-s + 0.856·23-s − 0.943·25-s + 0.140·27-s − 1.64·29-s − 0.918·31-s − 1.82·33-s − 0.0320·35-s + 0.890·37-s − 1.49·39-s + 1.84·41-s − 0.102·43-s + 0.261·45-s − 0.762·47-s − 0.981·49-s − 2.26·51-s − 1.36·53-s − 0.301·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 3 | \( 1 - 2.50T + 3T^{2} \) |
| 5 | \( 1 - 0.533T + 5T^{2} \) |
| 7 | \( 1 + 0.354T + 7T^{2} \) |
| 11 | \( 1 + 4.18T + 11T^{2} \) |
| 13 | \( 1 + 3.72T + 13T^{2} \) |
| 17 | \( 1 + 6.46T + 17T^{2} \) |
| 19 | \( 1 - 1.31T + 19T^{2} \) |
| 23 | \( 1 - 4.10T + 23T^{2} \) |
| 29 | \( 1 + 8.85T + 29T^{2} \) |
| 31 | \( 1 + 5.11T + 31T^{2} \) |
| 37 | \( 1 - 5.41T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 + 0.673T + 43T^{2} \) |
| 47 | \( 1 + 5.22T + 47T^{2} \) |
| 53 | \( 1 + 9.92T + 53T^{2} \) |
| 59 | \( 1 - 1.23T + 59T^{2} \) |
| 61 | \( 1 - 4.04T + 61T^{2} \) |
| 67 | \( 1 - 14.0T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 - 7.76T + 73T^{2} \) |
| 79 | \( 1 - 1.17T + 79T^{2} \) |
| 83 | \( 1 + 6.25T + 83T^{2} \) |
| 89 | \( 1 - 3.80T + 89T^{2} \) |
| 97 | \( 1 + 9.91T + 97T^{2} \) |
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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
−8.006053513229829694826185025691, −7.61446797821288947961012406060, −6.93544024226859832777793690650, −5.83471507498936726325952602553, −5.00343166108152345915794337094, −4.17070424436565861878954259536, −3.21876665733482097027510158639, −2.45319269308111346817202783528, −1.95250967370807271707688542593, 0,
1.95250967370807271707688542593, 2.45319269308111346817202783528, 3.21876665733482097027510158639, 4.17070424436565861878954259536, 5.00343166108152345915794337094, 5.83471507498936726325952602553, 6.93544024226859832777793690650, 7.61446797821288947961012406060, 8.006053513229829694826185025691
