| L(s) = 1 | − 2.44·2-s + 3.99·4-s − 2.44·5-s − 4.89·8-s + 5.99·10-s − 4.89·11-s − 4·13-s + 3.99·16-s + 2.44·17-s − 19-s − 9.79·20-s + 11.9·22-s − 2.44·23-s + 0.999·25-s + 9.79·26-s + 7.34·29-s − 7·31-s − 5.99·34-s + 8·37-s + 2.44·38-s + 11.9·40-s − 7.34·41-s − 43-s − 19.5·44-s + 5.99·46-s − 2.44·47-s − 2.44·50-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 1.99·4-s − 1.09·5-s − 1.73·8-s + 1.89·10-s − 1.47·11-s − 1.10·13-s + 0.999·16-s + 0.594·17-s − 0.229·19-s − 2.19·20-s + 2.55·22-s − 0.510·23-s + 0.199·25-s + 1.92·26-s + 1.36·29-s − 1.25·31-s − 1.02·34-s + 1.31·37-s + 0.397·38-s + 1.89·40-s − 1.14·41-s − 0.152·43-s − 2.95·44-s + 0.884·46-s − 0.357·47-s − 0.346·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2880983944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2880983944\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 5 | \( 1 + 2.44T + 5T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 2.44T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + 2.44T + 23T^{2} \) |
| 29 | \( 1 - 7.34T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + 7.34T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + 2.44T + 47T^{2} \) |
| 53 | \( 1 - 2.44T + 53T^{2} \) |
| 59 | \( 1 - 9.79T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 + 2.44T + 89T^{2} \) |
| 97 | \( 1 + T + 97T^{2} \) |
| 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
−9.849125313951014007785093658268, −8.581928299849798844515665283663, −8.105138752059572646524517237748, −7.50585473311539148064127254111, −6.92468644493538947964879493021, −5.60689318223731086794296073385, −4.50834285369947658073353825069, −3.10103467475367051351046320982, −2.14379640500151833657949044127, −0.47867821221014889538509864765,
0.47867821221014889538509864765, 2.14379640500151833657949044127, 3.10103467475367051351046320982, 4.50834285369947658073353825069, 5.60689318223731086794296073385, 6.92468644493538947964879493021, 7.50585473311539148064127254111, 8.105138752059572646524517237748, 8.581928299849798844515665283663, 9.849125313951014007785093658268
