| L(s) = 1 | − 5-s + 4·11-s − 2·13-s − 2·17-s + 19-s − 4·23-s + 25-s + 6·29-s + 4·31-s + 6·37-s − 10·41-s + 4·43-s + 12·47-s − 7·49-s + 6·53-s − 4·55-s − 12·59-s + 2·61-s + 2·65-s − 4·67-s − 8·71-s − 6·73-s − 4·79-s − 12·83-s + 2·85-s − 10·89-s − 95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.229·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s + 0.718·31-s + 0.986·37-s − 1.56·41-s + 0.609·43-s + 1.75·47-s − 49-s + 0.824·53-s − 0.539·55-s − 1.56·59-s + 0.256·61-s + 0.248·65-s − 0.488·67-s − 0.949·71-s − 0.702·73-s − 0.450·79-s − 1.31·83-s + 0.216·85-s − 1.05·89-s − 0.102·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 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
−14.59799812393202, −14.16063224955959, −13.79348290985522, −13.16413439686668, −12.50457943380781, −11.98937621001524, −11.77366484953007, −11.22659650641916, −10.50450642811017, −10.04474804085904, −9.542827314837588, −8.865881153313701, −8.534667838684895, −7.867344873890879, −7.274341374946000, −6.817980043002417, −6.180067022887141, −5.760878418420582, −4.780179245151711, −4.411319723715993, −3.908952295129764, −3.100178952828193, −2.547751863764116, −1.660201207156449, −0.9604257130722163, 0,
0.9604257130722163, 1.660201207156449, 2.547751863764116, 3.100178952828193, 3.908952295129764, 4.411319723715993, 4.780179245151711, 5.760878418420582, 6.180067022887141, 6.817980043002417, 7.274341374946000, 7.867344873890879, 8.534667838684895, 8.865881153313701, 9.542827314837588, 10.04474804085904, 10.50450642811017, 11.22659650641916, 11.77366484953007, 11.98937621001524, 12.50457943380781, 13.16413439686668, 13.79348290985522, 14.16063224955959, 14.59799812393202
