L(s) = 1 | + 2-s − 4-s + 2·5-s − 3·8-s + 2·10-s + 11-s + 13-s − 16-s + 6·17-s − 4·19-s − 2·20-s + 22-s + 8·23-s − 25-s + 26-s + 10·29-s + 5·32-s + 6·34-s + 6·37-s − 4·38-s − 6·40-s − 10·41-s + 4·43-s − 44-s + 8·46-s − 8·47-s − 7·49-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.894·5-s − 1.06·8-s + 0.632·10-s + 0.301·11-s + 0.277·13-s − 1/4·16-s + 1.45·17-s − 0.917·19-s − 0.447·20-s + 0.213·22-s + 1.66·23-s − 1/5·25-s + 0.196·26-s + 1.85·29-s + 0.883·32-s + 1.02·34-s + 0.986·37-s − 0.648·38-s − 0.948·40-s − 1.56·41-s + 0.609·43-s − 0.150·44-s + 1.17·46-s − 1.16·47-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.422614242\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.422614242\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.828201966522165801644231083131, −8.809781988911382660911278540879, −8.279842457480330856862908947423, −6.91274116401925090743914373893, −6.14885963657490762630313444248, −5.39216574159615869785069323377, −4.65912153519243395995881297600, −3.58566955557413367386868416593, −2.66792787949502047211352973601, −1.12169940393228089488805712125,
1.12169940393228089488805712125, 2.66792787949502047211352973601, 3.58566955557413367386868416593, 4.65912153519243395995881297600, 5.39216574159615869785069323377, 6.14885963657490762630313444248, 6.91274116401925090743914373893, 8.279842457480330856862908947423, 8.809781988911382660911278540879, 9.828201966522165801644231083131