| L(s) = 1 | − 2.56·3-s + 5-s − 0.561·7-s + 3.56·9-s − 11-s − 5.12·13-s − 2.56·15-s + 1.43·17-s − 6.56·19-s + 1.43·21-s − 1.12·23-s + 25-s − 1.43·27-s + 4.56·29-s − 3.68·31-s + 2.56·33-s − 0.561·35-s + 10.8·37-s + 13.1·39-s − 10·41-s + 3.12·43-s + 3.56·45-s + 1.12·47-s − 6.68·49-s − 3.68·51-s + 8.56·53-s − 55-s + ⋯ |
| L(s) = 1 | − 1.47·3-s + 0.447·5-s − 0.212·7-s + 1.18·9-s − 0.301·11-s − 1.42·13-s − 0.661·15-s + 0.348·17-s − 1.50·19-s + 0.313·21-s − 0.234·23-s + 0.200·25-s − 0.276·27-s + 0.847·29-s − 0.661·31-s + 0.445·33-s − 0.0949·35-s + 1.77·37-s + 2.10·39-s − 1.56·41-s + 0.476·43-s + 0.530·45-s + 0.163·47-s − 0.954·49-s − 0.515·51-s + 1.17·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6646446631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6646446631\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 7 | \( 1 + 0.561T + 7T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 - 1.43T + 17T^{2} \) |
| 19 | \( 1 + 6.56T + 19T^{2} \) |
| 23 | \( 1 + 1.12T + 23T^{2} \) |
| 29 | \( 1 - 4.56T + 29T^{2} \) |
| 31 | \( 1 + 3.68T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 3.12T + 43T^{2} \) |
| 47 | \( 1 - 1.12T + 47T^{2} \) |
| 53 | \( 1 - 8.56T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 0.561T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 6.56T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 9.12T + 79T^{2} \) |
| 83 | \( 1 + 10T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 8.87T + 97T^{2} \) |
| 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
−8.559202414867928651069858668655, −7.67117449863213213536049130570, −6.84169358848669726240826928162, −6.27886466657804272403601511331, −5.60444801949900884358286612260, −4.88035638560409393146406747047, −4.29590352062415944696367251610, −2.88726716973045666195765820789, −1.91029933032961674184459126658, −0.50005541082092645207911075490,
0.50005541082092645207911075490, 1.91029933032961674184459126658, 2.88726716973045666195765820789, 4.29590352062415944696367251610, 4.88035638560409393146406747047, 5.60444801949900884358286612260, 6.27886466657804272403601511331, 6.84169358848669726240826928162, 7.67117449863213213536049130570, 8.559202414867928651069858668655
