| L(s) = 1 | − 2.23·3-s + 2.00·9-s − 2.23·11-s + 4·13-s − 3·17-s − 2.23·19-s − 8.94·23-s + 2.23·27-s − 4·29-s − 8.94·31-s + 5.00·33-s + 8·37-s − 8.94·39-s + 5·41-s + 8.94·43-s + 8.94·47-s − 7·49-s + 6.70·51-s + 4·53-s + 5.00·57-s + 8.94·59-s − 8·61-s − 6.70·67-s + 20.0·69-s − 8.94·71-s − 9·73-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.29·3-s + 0.666·9-s − 0.674·11-s + 1.10·13-s − 0.727·17-s − 0.512·19-s − 1.86·23-s + 0.430·27-s − 0.742·29-s − 1.60·31-s + 0.870·33-s + 1.31·37-s − 1.43·39-s + 0.780·41-s + 1.36·43-s + 1.30·47-s − 49-s + 0.939·51-s + 0.549·53-s + 0.662·57-s + 1.16·59-s − 1.02·61-s − 0.819·67-s + 2.40·69-s − 1.06·71-s − 1.05·73-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6700048834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6700048834\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.23T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 23 | \( 1 + 8.94T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 8.94T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 - 8.94T + 43T^{2} \) |
| 47 | \( 1 - 8.94T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 6.70T + 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 + 9T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 6.70T + 83T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
| 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
−7.87947007029256221948434742723, −7.30177669724537553505152341655, −6.30048662820212128597592051853, −5.91076880022242422181209497171, −5.45350957374437843672198243034, −4.34586177095589575416896267621, −3.95778197531840804477655444328, −2.65467377156982528207715161233, −1.69816137559948652563091035093, −0.44863285696777878211461281221,
0.44863285696777878211461281221, 1.69816137559948652563091035093, 2.65467377156982528207715161233, 3.95778197531840804477655444328, 4.34586177095589575416896267621, 5.45350957374437843672198243034, 5.91076880022242422181209497171, 6.30048662820212128597592051853, 7.30177669724537553505152341655, 7.87947007029256221948434742723
