| L(s) = 1 | − 3.44·3-s + 2.91·5-s − 3.05·7-s + 8.90·9-s − 0.926·11-s + 0.486·13-s − 10.0·15-s − 17-s − 1.91·19-s + 10.5·21-s + 6.03·23-s + 3.50·25-s − 20.3·27-s − 4.39·29-s + 7.75·31-s + 3.19·33-s − 8.91·35-s − 10.3·37-s − 1.67·39-s + 10.1·41-s − 7.87·43-s + 25.9·45-s + 1.62·47-s + 2.33·49-s + 3.44·51-s + 11.4·53-s − 2.70·55-s + ⋯ |
| L(s) = 1 | − 1.99·3-s + 1.30·5-s − 1.15·7-s + 2.96·9-s − 0.279·11-s + 0.134·13-s − 2.59·15-s − 0.242·17-s − 0.440·19-s + 2.30·21-s + 1.25·23-s + 0.701·25-s − 3.91·27-s − 0.816·29-s + 1.39·31-s + 0.556·33-s − 1.50·35-s − 1.70·37-s − 0.268·39-s + 1.58·41-s − 1.20·43-s + 3.87·45-s + 0.237·47-s + 0.334·49-s + 0.483·51-s + 1.57·53-s − 0.364·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9074922791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9074922791\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 3.44T + 3T^{2} \) |
| 5 | \( 1 - 2.91T + 5T^{2} \) |
| 7 | \( 1 + 3.05T + 7T^{2} \) |
| 11 | \( 1 + 0.926T + 11T^{2} \) |
| 13 | \( 1 - 0.486T + 13T^{2} \) |
| 19 | \( 1 + 1.91T + 19T^{2} \) |
| 23 | \( 1 - 6.03T + 23T^{2} \) |
| 29 | \( 1 + 4.39T + 29T^{2} \) |
| 31 | \( 1 - 7.75T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 7.87T + 43T^{2} \) |
| 47 | \( 1 - 1.62T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 61 | \( 1 - 5.75T + 61T^{2} \) |
| 67 | \( 1 + 2.33T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 4.08T + 79T^{2} \) |
| 83 | \( 1 - 9.70T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 - 4.28T + 97T^{2} \) |
| 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
−7.40656065835753079423336436554, −6.68555910002372184371141616561, −6.49324361147095440033465828760, −5.69273156359525976763931037118, −5.36216999866270628434849200544, −4.58652138523485775812813448823, −3.67197458451323052164213765905, −2.49852920178290131977034798060, −1.51053450407661470457734758127, −0.54402827375575738531180572241,
0.54402827375575738531180572241, 1.51053450407661470457734758127, 2.49852920178290131977034798060, 3.67197458451323052164213765905, 4.58652138523485775812813448823, 5.36216999866270628434849200544, 5.69273156359525976763931037118, 6.49324361147095440033465828760, 6.68555910002372184371141616561, 7.40656065835753079423336436554
