| L(s) = 1 | + 3.31·3-s − 4.86·7-s + 8.02·9-s − 4.15·11-s − 2.66·13-s + 2.86·17-s − 0.651·19-s − 16.1·21-s + 4.65·23-s + 16.6·27-s + 29-s − 4.17·31-s − 13.7·33-s − 11.3·37-s − 8.85·39-s + 3.83·41-s − 12.0·43-s − 7.48·47-s + 16.7·49-s + 9.52·51-s − 1.61·53-s − 2.16·57-s − 3.33·59-s − 11.0·61-s − 39.0·63-s + 6.18·67-s + 15.4·69-s + ⋯ |
| L(s) = 1 | + 1.91·3-s − 1.84·7-s + 2.67·9-s − 1.25·11-s − 0.739·13-s + 0.695·17-s − 0.149·19-s − 3.52·21-s + 0.970·23-s + 3.20·27-s + 0.185·29-s − 0.749·31-s − 2.39·33-s − 1.86·37-s − 1.41·39-s + 0.599·41-s − 1.83·43-s − 1.09·47-s + 2.38·49-s + 1.33·51-s − 0.222·53-s − 0.286·57-s − 0.433·59-s − 1.41·61-s − 4.92·63-s + 0.755·67-s + 1.86·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 3.31T + 3T^{2} \) |
| 7 | \( 1 + 4.86T + 7T^{2} \) |
| 11 | \( 1 + 4.15T + 11T^{2} \) |
| 13 | \( 1 + 2.66T + 13T^{2} \) |
| 17 | \( 1 - 2.86T + 17T^{2} \) |
| 19 | \( 1 + 0.651T + 19T^{2} \) |
| 23 | \( 1 - 4.65T + 23T^{2} \) |
| 31 | \( 1 + 4.17T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 - 3.83T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 7.48T + 47T^{2} \) |
| 53 | \( 1 + 1.61T + 53T^{2} \) |
| 59 | \( 1 + 3.33T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 6.18T + 67T^{2} \) |
| 71 | \( 1 - 0.903T + 71T^{2} \) |
| 73 | \( 1 + 5.03T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 6.22T + 83T^{2} \) |
| 89 | \( 1 - 4.63T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
| 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
−7.83943269596177583159038248501, −7.08231642676026943320235382796, −6.77401158924232704923921018269, −5.54455529607438427497438484681, −4.69284778330385835830580179090, −3.62196405070431286654608507479, −3.09186136797709846047149744028, −2.73209657173301037212122394170, −1.67847732970190104608965525501, 0,
1.67847732970190104608965525501, 2.73209657173301037212122394170, 3.09186136797709846047149744028, 3.62196405070431286654608507479, 4.69284778330385835830580179090, 5.54455529607438427497438484681, 6.77401158924232704923921018269, 7.08231642676026943320235382796, 7.83943269596177583159038248501
