L(s) = 1 | − 24·2-s − 1.47e3·4-s + 1.67e4·7-s + 8.44e4·8-s − 5.34e5·11-s + 5.77e5·13-s − 4.01e5·14-s + 9.87e5·16-s − 6.90e6·17-s + 1.06e7·19-s + 1.28e7·22-s + 1.86e7·23-s − 1.38e7·26-s − 2.46e7·28-s − 1.28e8·29-s − 5.28e7·31-s − 1.96e8·32-s + 1.65e8·34-s + 1.82e8·37-s − 2.55e8·38-s − 3.08e8·41-s + 1.71e7·43-s + 7.86e8·44-s − 4.47e8·46-s + 2.68e9·47-s − 1.69e9·49-s − 8.50e8·52-s + ⋯ |
L(s) = 1 | − 0.530·2-s − 0.718·4-s + 0.376·7-s + 0.911·8-s − 1.00·11-s + 0.431·13-s − 0.199·14-s + 0.235·16-s − 1.17·17-s + 0.987·19-s + 0.530·22-s + 0.603·23-s − 0.228·26-s − 0.270·28-s − 1.16·29-s − 0.331·31-s − 1.03·32-s + 0.625·34-s + 0.431·37-s − 0.523·38-s − 0.415·41-s + 0.0177·43-s + 0.719·44-s − 0.320·46-s + 1.70·47-s − 0.858·49-s − 0.310·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 3 p^{3} T + p^{11} T^{2} \) |
| 7 | \( 1 - 2392 p T + p^{11} T^{2} \) |
| 11 | \( 1 + 534612 T + p^{11} T^{2} \) |
| 13 | \( 1 - 577738 T + p^{11} T^{2} \) |
| 17 | \( 1 + 6905934 T + p^{11} T^{2} \) |
| 19 | \( 1 - 10661420 T + p^{11} T^{2} \) |
| 23 | \( 1 - 18643272 T + p^{11} T^{2} \) |
| 29 | \( 1 + 128406630 T + p^{11} T^{2} \) |
| 31 | \( 1 + 52843168 T + p^{11} T^{2} \) |
| 37 | \( 1 - 182213314 T + p^{11} T^{2} \) |
| 41 | \( 1 + 308120442 T + p^{11} T^{2} \) |
| 43 | \( 1 - 17125708 T + p^{11} T^{2} \) |
| 47 | \( 1 - 2687348496 T + p^{11} T^{2} \) |
| 53 | \( 1 + 1596055698 T + p^{11} T^{2} \) |
| 59 | \( 1 - 5189203740 T + p^{11} T^{2} \) |
| 61 | \( 1 - 6956478662 T + p^{11} T^{2} \) |
| 67 | \( 1 - 15481826884 T + p^{11} T^{2} \) |
| 71 | \( 1 + 9791485272 T + p^{11} T^{2} \) |
| 73 | \( 1 + 1463791322 T + p^{11} T^{2} \) |
| 79 | \( 1 - 38116845680 T + p^{11} T^{2} \) |
| 83 | \( 1 + 29335099668 T + p^{11} T^{2} \) |
| 89 | \( 1 - 24992917110 T + p^{11} T^{2} \) |
| 97 | \( 1 + 75013568546 T + p^{11} T^{2} \) |
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
−9.662572762095747143132780674942, −8.843852527746543530606281541160, −7.991682645195826562615795853972, −7.11934858179432313599510331027, −5.57944796469263924601830309439, −4.77376602138295544567223008596, −3.64430619700881191000754583992, −2.22421868028396419396015970050, −1.00955341940831894074426466763, 0,
1.00955341940831894074426466763, 2.22421868028396419396015970050, 3.64430619700881191000754583992, 4.77376602138295544567223008596, 5.57944796469263924601830309439, 7.11934858179432313599510331027, 7.991682645195826562615795853972, 8.843852527746543530606281541160, 9.662572762095747143132780674942