| L(s) = 1 | + 2.29·3-s − 5-s − 3.88·7-s + 2.26·9-s − 6.30·11-s + 1.31·13-s − 2.29·15-s + 6.44·17-s − 4.87·19-s − 8.92·21-s + 3.57·23-s + 25-s − 1.68·27-s − 2.12·29-s + 4.55·31-s − 14.4·33-s + 3.88·35-s − 5.47·37-s + 3.02·39-s − 10.1·41-s + 6.06·43-s − 2.26·45-s − 1.28·47-s + 8.13·49-s + 14.7·51-s + 7.12·53-s + 6.30·55-s + ⋯ |
| L(s) = 1 | + 1.32·3-s − 0.447·5-s − 1.47·7-s + 0.755·9-s − 1.90·11-s + 0.366·13-s − 0.592·15-s + 1.56·17-s − 1.11·19-s − 1.94·21-s + 0.746·23-s + 0.200·25-s − 0.324·27-s − 0.394·29-s + 0.817·31-s − 2.52·33-s + 0.657·35-s − 0.899·37-s + 0.484·39-s − 1.58·41-s + 0.925·43-s − 0.337·45-s − 0.187·47-s + 1.16·49-s + 2.07·51-s + 0.978·53-s + 0.850·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.769389791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.769389791\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 - 2.29T + 3T^{2} \) |
| 7 | \( 1 + 3.88T + 7T^{2} \) |
| 11 | \( 1 + 6.30T + 11T^{2} \) |
| 13 | \( 1 - 1.31T + 13T^{2} \) |
| 17 | \( 1 - 6.44T + 17T^{2} \) |
| 19 | \( 1 + 4.87T + 19T^{2} \) |
| 23 | \( 1 - 3.57T + 23T^{2} \) |
| 29 | \( 1 + 2.12T + 29T^{2} \) |
| 31 | \( 1 - 4.55T + 31T^{2} \) |
| 37 | \( 1 + 5.47T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 6.06T + 43T^{2} \) |
| 47 | \( 1 + 1.28T + 47T^{2} \) |
| 53 | \( 1 - 7.12T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 + 7.40T + 67T^{2} \) |
| 71 | \( 1 + 5.50T + 71T^{2} \) |
| 73 | \( 1 - 8.15T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 0.147T + 89T^{2} \) |
| 97 | \( 1 - 1.77T + 97T^{2} \) |
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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
−7.993598266629088818716638075541, −7.28696024986006391380765365192, −6.65692636210990049945096757446, −5.69534761182060080935658890433, −5.08320690621034074426484279923, −3.88331659480580876274539717753, −3.36932309201109613444989205350, −2.83856554001039942106882751986, −2.15078882251656209041327498273, −0.57837762276197763710751715188,
0.57837762276197763710751715188, 2.15078882251656209041327498273, 2.83856554001039942106882751986, 3.36932309201109613444989205350, 3.88331659480580876274539717753, 5.08320690621034074426484279923, 5.69534761182060080935658890433, 6.65692636210990049945096757446, 7.28696024986006391380765365192, 7.993598266629088818716638075541
