| L(s) = 1 | − 3-s − 5-s − 4·7-s + 9-s + 4·11-s + 6·13-s + 15-s + 2·17-s + 4·19-s + 4·21-s + 25-s − 27-s + 10·29-s − 4·31-s − 4·33-s + 4·35-s − 10·37-s − 6·39-s + 2·41-s − 4·43-s − 45-s + 8·47-s + 9·49-s − 2·51-s + 2·53-s − 4·55-s − 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s + 0.258·15-s + 0.485·17-s + 0.917·19-s + 0.872·21-s + 1/5·25-s − 0.192·27-s + 1.85·29-s − 0.718·31-s − 0.696·33-s + 0.676·35-s − 1.64·37-s − 0.960·39-s + 0.312·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s + 9/7·49-s − 0.280·51-s + 0.274·53-s − 0.539·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.046977174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.046977174\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−11.02123424258128403111626112733, −10.13855312007199582968173453548, −9.265808981321240617130460115923, −8.421933311737600437687566687322, −6.97981434955619225320560223720, −6.46117181371757757685905507069, −5.52040688744560586491461101623, −3.95366310025504614794426142117, −3.30373640608268734535981700762, −1.02432023626640801365429192064,
1.02432023626640801365429192064, 3.30373640608268734535981700762, 3.95366310025504614794426142117, 5.52040688744560586491461101623, 6.46117181371757757685905507069, 6.97981434955619225320560223720, 8.421933311737600437687566687322, 9.265808981321240617130460115923, 10.13855312007199582968173453548, 11.02123424258128403111626112733
