L(s) = 1 | − 2.37·3-s − 0.155·5-s − 7-s + 2.64·9-s + 11-s + 13-s + 0.369·15-s − 3.88·17-s − 3.56·19-s + 2.37·21-s + 5.09·23-s − 4.97·25-s + 0.849·27-s − 6.55·29-s + 4.73·31-s − 2.37·33-s + 0.155·35-s + 7.33·37-s − 2.37·39-s + 11.7·41-s + 7.29·43-s − 0.411·45-s − 9.56·47-s + 49-s + 9.23·51-s − 8.39·53-s − 0.155·55-s + ⋯ |
L(s) = 1 | − 1.37·3-s − 0.0696·5-s − 0.377·7-s + 0.880·9-s + 0.301·11-s + 0.277·13-s + 0.0954·15-s − 0.943·17-s − 0.818·19-s + 0.518·21-s + 1.06·23-s − 0.995·25-s + 0.163·27-s − 1.21·29-s + 0.850·31-s − 0.413·33-s + 0.0263·35-s + 1.20·37-s − 0.380·39-s + 1.84·41-s + 1.11·43-s − 0.0613·45-s − 1.39·47-s + 0.142·49-s + 1.29·51-s − 1.15·53-s − 0.0209·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + 2.37T + 3T^{2} \) |
| 5 | \( 1 + 0.155T + 5T^{2} \) |
| 17 | \( 1 + 3.88T + 17T^{2} \) |
| 19 | \( 1 + 3.56T + 19T^{2} \) |
| 23 | \( 1 - 5.09T + 23T^{2} \) |
| 29 | \( 1 + 6.55T + 29T^{2} \) |
| 31 | \( 1 - 4.73T + 31T^{2} \) |
| 37 | \( 1 - 7.33T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 7.29T + 43T^{2} \) |
| 47 | \( 1 + 9.56T + 47T^{2} \) |
| 53 | \( 1 + 8.39T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 8.66T + 61T^{2} \) |
| 67 | \( 1 + 9.49T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 1.56T + 73T^{2} \) |
| 79 | \( 1 + 3.14T + 79T^{2} \) |
| 83 | \( 1 - 7.49T + 83T^{2} \) |
| 89 | \( 1 - 0.972T + 89T^{2} \) |
| 97 | \( 1 + 4.23T + 97T^{2} \) |
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
−7.32322072957569979169435867200, −6.56677555745726382600595497198, −6.09507726461805120432265097810, −5.62583138175127792569438768058, −4.55019991270548767022584369218, −4.27232550779958299645048135170, −3.13710020209251741921458513622, −2.12618314041601491093148987829, −0.971064677565740629932276614424, 0,
0.971064677565740629932276614424, 2.12618314041601491093148987829, 3.13710020209251741921458513622, 4.27232550779958299645048135170, 4.55019991270548767022584369218, 5.62583138175127792569438768058, 6.09507726461805120432265097810, 6.56677555745726382600595497198, 7.32322072957569979169435867200