| L(s) = 1 | + 2.89·3-s − 0.776·5-s − 7-s + 5.35·9-s + 4.69·11-s + 0.513·13-s − 2.24·15-s + 5.71·17-s − 3.33·19-s − 2.89·21-s + 23-s − 4.39·25-s + 6.82·27-s + 3.51·29-s − 0.262·31-s + 13.5·33-s + 0.776·35-s + 2.62·37-s + 1.48·39-s − 0.912·41-s + 7.73·43-s − 4.15·45-s − 0.891·47-s + 49-s + 16.5·51-s − 13.5·53-s − 3.64·55-s + ⋯ |
| L(s) = 1 | + 1.66·3-s − 0.347·5-s − 0.377·7-s + 1.78·9-s + 1.41·11-s + 0.142·13-s − 0.579·15-s + 1.38·17-s − 0.765·19-s − 0.630·21-s + 0.208·23-s − 0.879·25-s + 1.31·27-s + 0.652·29-s − 0.0470·31-s + 2.36·33-s + 0.131·35-s + 0.432·37-s + 0.237·39-s − 0.142·41-s + 1.17·43-s − 0.620·45-s − 0.129·47-s + 0.142·49-s + 2.31·51-s − 1.86·53-s − 0.491·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.446454074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.446454074\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2.89T + 3T^{2} \) |
| 5 | \( 1 + 0.776T + 5T^{2} \) |
| 11 | \( 1 - 4.69T + 11T^{2} \) |
| 13 | \( 1 - 0.513T + 13T^{2} \) |
| 17 | \( 1 - 5.71T + 17T^{2} \) |
| 19 | \( 1 + 3.33T + 19T^{2} \) |
| 29 | \( 1 - 3.51T + 29T^{2} \) |
| 31 | \( 1 + 0.262T + 31T^{2} \) |
| 37 | \( 1 - 2.62T + 37T^{2} \) |
| 41 | \( 1 + 0.912T + 41T^{2} \) |
| 43 | \( 1 - 7.73T + 43T^{2} \) |
| 47 | \( 1 + 0.891T + 47T^{2} \) |
| 53 | \( 1 + 13.5T + 53T^{2} \) |
| 59 | \( 1 - 4.37T + 59T^{2} \) |
| 61 | \( 1 + 1.40T + 61T^{2} \) |
| 67 | \( 1 + 5.17T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 + 1.89T + 73T^{2} \) |
| 79 | \( 1 + 4.69T + 79T^{2} \) |
| 83 | \( 1 - 18.0T + 83T^{2} \) |
| 89 | \( 1 - 8.78T + 89T^{2} \) |
| 97 | \( 1 - 8.11T + 97T^{2} \) |
| 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
−8.927966698020279829381950688481, −8.107279604485108458956073926547, −7.65277188436756878702200416399, −6.73992291033183208365316793906, −5.99154797874231138795794562105, −4.58710643064433910929059030320, −3.75954695328496757241650377084, −3.32986577141801782469196947985, −2.26450022920236287040050020107, −1.19382451256770612638820605574,
1.19382451256770612638820605574, 2.26450022920236287040050020107, 3.32986577141801782469196947985, 3.75954695328496757241650377084, 4.58710643064433910929059030320, 5.99154797874231138795794562105, 6.73992291033183208365316793906, 7.65277188436756878702200416399, 8.107279604485108458956073926547, 8.927966698020279829381950688481
