L(s) = 1 | + 3-s − 5-s + 9-s − 4·11-s + 2·13-s − 15-s − 2·17-s + 4·19-s − 8·23-s + 25-s + 27-s + 2·29-s + 8·31-s − 4·33-s + 2·39-s + 10·41-s + 12·43-s − 45-s − 7·49-s − 2·51-s + 6·53-s + 4·55-s + 4·57-s + 4·59-s + 10·61-s − 2·65-s + 4·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.258·15-s − 0.485·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.696·33-s + 0.320·39-s + 1.56·41-s + 1.82·43-s − 0.149·45-s − 49-s − 0.280·51-s + 0.824·53-s + 0.539·55-s + 0.529·57-s + 0.520·59-s + 1.28·61-s − 0.248·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.83729933908720, −12.32881898966781, −12.08862664811675, −11.27923341428237, −11.16829677681084, −10.51150159766073, −9.981237266075078, −9.788699176507728, −9.140732616595835, −8.543793080761388, −8.224742418776649, −7.836788730470677, −7.439816192455709, −6.881789034190549, −6.327047725977930, −5.673632183633607, −5.461450931503454, −4.589704928429461, −4.202521739286726, −3.862128801733935, −3.046855839378173, −2.632034010686063, −2.288381400154437, −1.376349213678937, −0.7980392206997696, 0,
0.7980392206997696, 1.376349213678937, 2.288381400154437, 2.632034010686063, 3.046855839378173, 3.862128801733935, 4.202521739286726, 4.589704928429461, 5.461450931503454, 5.673632183633607, 6.327047725977930, 6.881789034190549, 7.439816192455709, 7.836788730470677, 8.224742418776649, 8.543793080761388, 9.140732616595835, 9.788699176507728, 9.981237266075078, 10.51150159766073, 11.16829677681084, 11.27923341428237, 12.08862664811675, 12.32881898966781, 12.83729933908720