L(s) = 1 | + 2.73·3-s + 5-s + 4.73·7-s + 4.46·9-s − 3.46·11-s + 3.46·13-s + 2.73·15-s − 3.46·17-s − 2·19-s + 12.9·21-s − 2.19·23-s + 25-s + 3.99·27-s + 2.53·31-s − 9.46·33-s + 4.73·35-s − 6·37-s + 9.46·39-s − 9.46·41-s + 0.196·43-s + 4.46·45-s − 2.19·47-s + 15.3·49-s − 9.46·51-s − 10.3·53-s − 3.46·55-s − 5.46·57-s + ⋯ |
L(s) = 1 | + 1.57·3-s + 0.447·5-s + 1.78·7-s + 1.48·9-s − 1.04·11-s + 0.960·13-s + 0.705·15-s − 0.840·17-s − 0.458·19-s + 2.82·21-s − 0.457·23-s + 0.200·25-s + 0.769·27-s + 0.455·31-s − 1.64·33-s + 0.799·35-s − 0.986·37-s + 1.51·39-s − 1.47·41-s + 0.0299·43-s + 0.665·45-s − 0.320·47-s + 2.19·49-s − 1.32·51-s − 1.42·53-s − 0.467·55-s − 0.723·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.510675861\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.510675861\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 7 | \( 1 - 4.73T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 2.19T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 2.53T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 9.46T + 41T^{2} \) |
| 43 | \( 1 - 0.196T + 43T^{2} \) |
| 47 | \( 1 + 2.19T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 0.928T + 61T^{2} \) |
| 67 | \( 1 + 0.196T + 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 + 6.39T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 1.26T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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
−9.445026292212794170261431296328, −8.564479835042351229881647927291, −8.259165428997083822799594819274, −7.63069463683182740283531031273, −6.49645254919945655010070597892, −5.22687861877540423997021405002, −4.48527093005597921647991269512, −3.40438788261367541822551840578, −2.24634652414582701977966595200, −1.66452138663424647605199742595,
1.66452138663424647605199742595, 2.24634652414582701977966595200, 3.40438788261367541822551840578, 4.48527093005597921647991269512, 5.22687861877540423997021405002, 6.49645254919945655010070597892, 7.63069463683182740283531031273, 8.259165428997083822799594819274, 8.564479835042351229881647927291, 9.445026292212794170261431296328