| L(s) = 1 | − 2·3-s + 8·7-s − 23·9-s − 11·11-s + 25·13-s − 52·17-s + 19·19-s − 16·21-s − 75·23-s + 100·27-s + 241·29-s + 189·31-s + 22·33-s + 68·37-s − 50·39-s + 240·41-s − 183·43-s − 56·47-s − 279·49-s + 104·51-s − 142·53-s − 38·57-s − 584·59-s − 622·61-s − 184·63-s − 142·67-s + 150·69-s + ⋯ |
| L(s) = 1 | − 0.384·3-s + 0.431·7-s − 0.851·9-s − 0.301·11-s + 0.533·13-s − 0.741·17-s + 0.229·19-s − 0.166·21-s − 0.679·23-s + 0.712·27-s + 1.54·29-s + 1.09·31-s + 0.116·33-s + 0.302·37-s − 0.205·39-s + 0.914·41-s − 0.649·43-s − 0.173·47-s − 0.813·49-s + 0.285·51-s − 0.368·53-s − 0.0883·57-s − 1.28·59-s − 1.30·61-s − 0.367·63-s − 0.258·67-s + 0.261·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + p T \) |
| good | 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 7 | \( 1 - 8 T + p^{3} T^{2} \) |
| 13 | \( 1 - 25 T + p^{3} T^{2} \) |
| 17 | \( 1 + 52 T + p^{3} T^{2} \) |
| 19 | \( 1 - p T + p^{3} T^{2} \) |
| 23 | \( 1 + 75 T + p^{3} T^{2} \) |
| 29 | \( 1 - 241 T + p^{3} T^{2} \) |
| 31 | \( 1 - 189 T + p^{3} T^{2} \) |
| 37 | \( 1 - 68 T + p^{3} T^{2} \) |
| 41 | \( 1 - 240 T + p^{3} T^{2} \) |
| 43 | \( 1 + 183 T + p^{3} T^{2} \) |
| 47 | \( 1 + 56 T + p^{3} T^{2} \) |
| 53 | \( 1 + 142 T + p^{3} T^{2} \) |
| 59 | \( 1 + 584 T + p^{3} T^{2} \) |
| 61 | \( 1 + 622 T + p^{3} T^{2} \) |
| 67 | \( 1 + 142 T + p^{3} T^{2} \) |
| 71 | \( 1 - 435 T + p^{3} T^{2} \) |
| 73 | \( 1 - 628 T + p^{3} T^{2} \) |
| 79 | \( 1 + 824 T + p^{3} T^{2} \) |
| 83 | \( 1 - 837 T + p^{3} T^{2} \) |
| 89 | \( 1 + 53 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1677 T + p^{3} T^{2} \) |
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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
−8.289916275398438453909158844789, −7.72048733595426276621122572778, −6.45313673394716925085904992544, −6.13391164986924104607627664331, −5.04318223553267794506510663749, −4.46708329959281275409805483568, −3.24435970901190822844355537329, −2.38110285637934953454618807911, −1.13797840030168251548035863843, 0,
1.13797840030168251548035863843, 2.38110285637934953454618807911, 3.24435970901190822844355537329, 4.46708329959281275409805483568, 5.04318223553267794506510663749, 6.13391164986924104607627664331, 6.45313673394716925085904992544, 7.72048733595426276621122572778, 8.289916275398438453909158844789
