| L(s) = 1 | − 2·5-s − 7-s + 4·17-s − 4·19-s + 4·23-s − 25-s + 2·29-s − 2·31-s + 2·35-s − 6·37-s + 4·41-s + 4·43-s − 2·47-s + 49-s − 2·53-s + 6·59-s − 4·61-s + 12·71-s − 16·73-s + 8·79-s − 12·83-s − 8·85-s − 10·89-s + 8·95-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s + 0.970·17-s − 0.917·19-s + 0.834·23-s − 1/5·25-s + 0.371·29-s − 0.359·31-s + 0.338·35-s − 0.986·37-s + 0.624·41-s + 0.609·43-s − 0.291·47-s + 1/7·49-s − 0.274·53-s + 0.781·59-s − 0.512·61-s + 1.42·71-s − 1.87·73-s + 0.900·79-s − 1.31·83-s − 0.867·85-s − 1.05·89-s + 0.820·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.57170930001103, −14.15088367947220, −13.48975751111556, −12.92774105319167, −12.49406573455349, −12.07440914206252, −11.57932068763039, −10.82073911819995, −10.74855582804051, −9.793893021175540, −9.599075214663343, −8.630208008894483, −8.504093387927544, −7.755313409021579, −7.251067750943152, −6.852136598104275, −6.069133346363136, −5.621948227072670, −4.874854376929819, −4.303959321939559, −3.706665062486619, −3.211324045769080, −2.540106762703376, −1.671385937642997, −0.8109239376834782, 0,
0.8109239376834782, 1.671385937642997, 2.540106762703376, 3.211324045769080, 3.706665062486619, 4.303959321939559, 4.874854376929819, 5.621948227072670, 6.069133346363136, 6.852136598104275, 7.251067750943152, 7.755313409021579, 8.504093387927544, 8.630208008894483, 9.599075214663343, 9.793893021175540, 10.74855582804051, 10.82073911819995, 11.57932068763039, 12.07440914206252, 12.49406573455349, 12.92774105319167, 13.48975751111556, 14.15088367947220, 14.57170930001103
