L(s) = 1 | − 8·4-s − 2·7-s + 11·11-s + 22·13-s + 64·16-s + 72·17-s + 122·19-s + 72·23-s + 16·28-s − 96·29-s − 112·31-s − 266·37-s + 96·41-s + 382·43-s − 88·44-s + 360·47-s − 339·49-s − 176·52-s + 318·53-s − 660·59-s − 430·61-s − 512·64-s − 380·67-s − 576·68-s − 168·71-s − 218·73-s − 976·76-s + ⋯ |
L(s) = 1 | − 4-s − 0.107·7-s + 0.301·11-s + 0.469·13-s + 16-s + 1.02·17-s + 1.47·19-s + 0.652·23-s + 0.107·28-s − 0.614·29-s − 0.648·31-s − 1.18·37-s + 0.365·41-s + 1.35·43-s − 0.301·44-s + 1.11·47-s − 0.988·49-s − 0.469·52-s + 0.824·53-s − 1.45·59-s − 0.902·61-s − 64-s − 0.692·67-s − 1.02·68-s − 0.280·71-s − 0.349·73-s − 1.47·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.851014407\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.851014407\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - p T \) |
good | 2 | \( 1 + p^{3} T^{2} \) |
| 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 13 | \( 1 - 22 T + p^{3} T^{2} \) |
| 17 | \( 1 - 72 T + p^{3} T^{2} \) |
| 19 | \( 1 - 122 T + p^{3} T^{2} \) |
| 23 | \( 1 - 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 96 T + p^{3} T^{2} \) |
| 31 | \( 1 + 112 T + p^{3} T^{2} \) |
| 37 | \( 1 + 266 T + p^{3} T^{2} \) |
| 41 | \( 1 - 96 T + p^{3} T^{2} \) |
| 43 | \( 1 - 382 T + p^{3} T^{2} \) |
| 47 | \( 1 - 360 T + p^{3} T^{2} \) |
| 53 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 59 | \( 1 + 660 T + p^{3} T^{2} \) |
| 61 | \( 1 + 430 T + p^{3} T^{2} \) |
| 67 | \( 1 + 380 T + p^{3} T^{2} \) |
| 71 | \( 1 + 168 T + p^{3} T^{2} \) |
| 73 | \( 1 + 218 T + p^{3} T^{2} \) |
| 79 | \( 1 + 706 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1068 T + p^{3} T^{2} \) |
| 89 | \( 1 - 6 T + p^{3} T^{2} \) |
| 97 | \( 1 + 686 T + p^{3} T^{2} \) |
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
−8.860872144013311557078311763388, −7.69712070944530303516112374812, −7.33891967863702780750979534766, −6.00835460031406068588026934835, −5.47286509023439640332969518284, −4.64315113111196158262657465848, −3.66218924489726904364231336149, −3.10726344841155881755694633463, −1.50030326201122809911606334371, −0.65882301726839860155247978156,
0.65882301726839860155247978156, 1.50030326201122809911606334371, 3.10726344841155881755694633463, 3.66218924489726904364231336149, 4.64315113111196158262657465848, 5.47286509023439640332969518284, 6.00835460031406068588026934835, 7.33891967863702780750979534766, 7.69712070944530303516112374812, 8.860872144013311557078311763388