| L(s) = 1 | + 2·3-s + 7-s + 9-s − 11-s − 4·13-s − 4·19-s + 2·21-s − 4·27-s + 6·29-s + 10·31-s − 2·33-s + 2·37-s − 8·39-s − 12·41-s + 4·43-s + 6·47-s + 49-s − 6·53-s − 8·57-s − 6·59-s + 4·61-s + 63-s + 4·67-s − 12·71-s + 4·73-s − 77-s − 8·79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 0.917·19-s + 0.436·21-s − 0.769·27-s + 1.11·29-s + 1.79·31-s − 0.348·33-s + 0.328·37-s − 1.28·39-s − 1.87·41-s + 0.609·43-s + 0.875·47-s + 1/7·49-s − 0.824·53-s − 1.05·57-s − 0.781·59-s + 0.512·61-s + 0.125·63-s + 0.488·67-s − 1.42·71-s + 0.468·73-s − 0.113·77-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.90172438609688, −13.37656564407438, −12.94318286453319, −12.33961170423526, −11.88491559385869, −11.50818539251766, −10.74038672629059, −10.19197339605910, −9.999599319920491, −9.293897841017308, −8.776273187712831, −8.404184843714448, −7.950557160671467, −7.546402856645763, −6.902151178361247, −6.393422052675083, −5.767248228657358, −5.039127804392132, −4.542333225949956, −4.183152215199532, −3.230914657165590, −2.907285295817406, −2.299791210633516, −1.864600676786978, −0.9231819059587231, 0,
0.9231819059587231, 1.864600676786978, 2.299791210633516, 2.907285295817406, 3.230914657165590, 4.183152215199532, 4.542333225949956, 5.039127804392132, 5.767248228657358, 6.393422052675083, 6.902151178361247, 7.546402856645763, 7.950557160671467, 8.404184843714448, 8.776273187712831, 9.293897841017308, 9.999599319920491, 10.19197339605910, 10.74038672629059, 11.50818539251766, 11.88491559385869, 12.33961170423526, 12.94318286453319, 13.37656564407438, 13.90172438609688
